And once you see this, you know you have chose to immerse yourself in the game that is Defense Of The Ancients 2 (or DOTA 2 for short).
Once upon a time ago, Warcraft 3 was modified in a manner that made its result more famous than Warcraft itself. Valve, being awesome, bought the license and is now developing DOTA 2 as its own Multiplayer Online Battle Arena (MOBA for short), a unique genre that mixes elements of Real-Time Strategy and RolePlaying Games in that you have to take over the map with allies (whether bots or friends) while using unique Heroes with very specific strengths.
The screen you see above you is the front page's News feed, with navigation and settings on the top plus community interaction on the right. Your wins will always be displayed publicly (while losses are private), and your friends and party/team based chat screens will always be in view until you head into a game.
Take a few clicks to get familiar with the layout of the Navigation bar. Particularly Settings (to which we HIGHLY suggest you take the time to turn off Auto-Attack), Play, and Learn.
Speaking of Learn, it's a good idea to try to familiarize yourself with what you're getting into, right?

Welp. Thus far, there is no proper tutorial mode for DOTA 2. The Learn menu, as you see, normally consists of a grid of each Hero's image and another tab up top that swaps you over to every item in the game. It's not a bad idea to click around and examine heroes and items; clicking a hero will send you into a page describing the hero's roles, stat upgrades (per point and level), and a detailed description of each of their "spells" (which you might think of as their skills, feats, perks, or powers).
First off, they are grouped under a few traits. Primary attribute (Strength, Agility, Intelligence), Role, and Attack Type. Primary Attribute, as you may expect, is what their attribute focus will be. Don't let the normal stereotypes fool you, Strength heroes are not always the heavy hitters and Intelligence heroes are not always the fancy spellsingers. If you've ever played Dynasty Warriors, you can compare its Officers to DOTA 2's heroes in how versatile they are in their specialized skills. Typically, for Attributes, expect the following.

Strength - High HP, good armor, strong blows. Often Durable, almost always Melee, often apply Spells that expose a single target to (powerful) basic attacks.
Agility - High attack speed, good movement speed, fast killer. Often Carry, almost equal Melee / Ranged, often isolates a single target to dispatch very quickly by slowing enemies or buffing one's self.
Intelligence - High mana, good attack, powerful spells. Often Support, almost always Ranged, extremely specialized Spell set that makes them disastrous to catch in certain situations.

"Role" is a term that's as important as you make it. Ursa, the example highlighted, reads "Carry", "Jungler", and "Durable". This means he's a team carrier when stronger and with good items, able to break off the normal fights to get strength and gold in the Jungle areas, and is a tougher hero to kill than normal when built up. However, save for some things like Disabler (which requires stunning, silencing, or such effects) and Support (which requires boosting the health or stats of allies), he can technically be played however you want. Roles are more often than not a suggestion of what to expect than a demand for how to play. They consist of the following.

Carry - If they get good levels and gold flow, they become nigh unstoppable in the late game.
Disabler - They can effectively remove a problem target from being able cause problems (often silencing, stunning, or transforming the enemy)
Durable - Often involves having Spells that make it much harder to hit or damage the Hero.
Escape - Able to get an advantage in escaping a suicidal situation, sometimes good at helping allies do the same during team fights.
Initiator - Good team fight starters, they very often have Spells that disrupt crowds with a status effect or cause environmental hazards.
Jungler - Can fight in the Jungles during the early game for an advantageous boost to Experience and Gold. (Requires a team capable of surviving numerous heroes)
Lane Support - Good at helping you control and secure a lane early in the game.
Nuker - Learns spells that deal immense amounts of damage in short times.
Pusher - Usually involves summoning minions / converting enemies to give a stronger push into enemy lines, particularly against towers.
Support - Heals allies, boosts stats, generally focusing on strengthening your teammates.

It's a lot to take in. But as stated above, Attributes more than anything dictate your item choices in game and Roles are suggestions to how to play the character. Find one whose spells look like they work to your advantage and pick them out to use a few times, if you want.

So, once you've found something you want to try out (for example, a Sniper with a goal of the Butterfly item for immense long-range havoc wreaking), next comes the Play screen. If you have friends you want to party with, clicking their icon will allow you to join together before trying to enter matches. Otherwise, you'll have to go online to join a server, find a Lobby hosted by friends, or host your own Lobby.
A "Lobby", as you might expect, is a game you make where it's essentially people you are inviting to a game specifically set up to your desires. Add bots, penalize a faction, enable cheats, or just play solo against creeps to explore the map and character.
For the sake of this tutorial, I will almost exclusively be demonstrating in a game where I play against one Passive bot.

One note that must be made before entering the games will be the Game Modes. "All Pick" and "Single Draft" are different modes of Hero selection.

All Pick - Lets you choose from the entire list of heroes. Conversely, this also means your allies and enemies choose from the same list of everyone. If you absolutely want one specific Hero but someone else chooses it, that's that. You have to go for another. It's worth noting that you can select "Random", which offers a 250 Gold boost to your character when entering the game. This is not an insignificant amount, as most early stage items can cost about that much. (The default amount is 600, so the boosted amount will be 850.)
Single Draft - The game chooses three heroes for you and you get to select one of the three. This prevents any possible overlap of people wanting a specific Hero, but can keep you from using a familiar face for quite some time. Single Draft is good for anyone who wants to learn as many different Heroes as possible, which is useful to anyone who wants to take advantage of the Random feature's early game gold boost in All Pick games.

As you can probably tell by the greyed out "Learn To Play" options, there's no proper way to learn to play without jumping in feet first. If you don't mind a less than perfect record, Find A Match. Otherwise, Create a Lobby and practice with friends versus bots. (Medium bots on all sides is a fair enough accurate recreation of a normal game's difficulty, assuming you are not playing any tournament-strength players.)

After you get through the loading screen for the server, you'll come across one that shows every player's status in connecting to the game. Make your pre-game banter (Enter to talk to allies, or Shift-Enter to talk to the entire game) or just pray nobody idles out of the game and forces you to find a new one while waiting on everyone to connect. Once they do, you'll see the above (in an All Pick setting). You can immediately type in the first few letters of a Hero you want to use (d-r-o to jump to Drow Ranger, for example), scroll manually through the heroes, or as pointed out earlier, use the "Pick Random Hero" button to take advantage of a Random selection's perks.

Once your hero is chosen, you can back out and pick another Hero. However, this gives you a 100 Gold penalty, removes the 250 gold boost to selecting a Random hero, and can only be done once a game. If you want to return to the Hero list to get a last second glance at your Hero's spells and roles, the "Return to Browsing" button will place you back at the selection screen where you can hover over these things to examine. Once you're set, click the "Play" button with your Hero's face.

It's not uncommon for some players to hold back picking their Hero until they know what the other Heroes in use are. Allies may do it to balance a team, or as a literal counter to a specific problem enemy Hero. As time passes, you'll get to know who does well against who.

And now you're confused. A lot of stuff around that's self explanitory if you have the time to look it all over, but since the game does not accommodate new players that well yet, you have about a minute to look over your new surroundings and get into position before the game gets going. So let's get the different things cleared up so you know what to worry about when.

1 - Back button. During the game, you're allowed to hide the game and return to the Main Menu, which lets you adjust settings or disconnect from the server.
2 - Game information. Contains chat buttons, hero scores, settings, and so on.
3 - Radiant Heroes. On the very left is "Sand King" (the character I Random rolled).
4 - Radiant kill count, Time clock (which does matter, since night time makes it harder to see and can boost Night Stalker's abilities), Dire kill count.
5 - Dire Heroes. On the very left is "Earthshaker", the hero Dire's bot selected.
6 - Your counts for Hero Kills, Deaths, Assists, Last Hits, and Denies. ("Deny" is a trick where you kill an allied unit to keep enemies from getting exp or gold.)
7 - Shop and Shopkeeper. You will always want to have an eye on what items you need to get to further advance your usefulness.
8 - Your Base area. This general area offers a high Health/Mana recovery rate, and contains a powerful construct that prevents spawn camping.
9 - Minimap. Radiant structures and Green, Dire structures are Red. Heroes will be marked by their slot color (comes in handy in 5 vs 5 matches). There will be dark green triangles that represent Jungle creatures (which are farmed for Gold and Experience when the game starts. The red triangular thing near the river is "Roshan", a powerful neutral beast that offers a good boost to whichever team kills it.
10 - Level Up your Hero. When you click Level Up, you can choose to boost one Spell (assuming you have not maximized its level and have passed the Milestone needed to boost it), or boost one of your three Attributes by +2 (almost always your Primary). Hover over the icons to find out your hero's current specifications.
11 - The green bar is health, and the blue bar is mana. (Who would have guessed?) The four squares below it are your Spells. A solid gold square will mean its level is upgraded. A pale, dim gold square means it can be upgraded upon level-up.
12 - Inventory. You always have six slots for items you can hold (though you have six Stash slots if you need to hold items for a while).
13 - "Glyph", which boosts the defence of your structures for a few seconds and has a five minute cooldown. Good if you or any ally just need a few seconds to arrive at a structure to save it.

So, to summarize what to do at this screen. Level up a Spell, decide who goes to what Lanes (top, middle, and bottom), and buy your items. Remember, left click is examine / select while right click is interact (movement, attacking, purchasing at a shop). Speaking of which...

The shop system in DOTA 2 is kind enough to offer you suggestions for items in all stages of the game based on your hero. Stat boosters, specific healing items, and so on will be more important to some than to others, and the makers try to recognize that with their suggestions.

Sand King in specific is a Strength hero who deals with rushing into battle quickly and dealing high damage against stunned enemies. So the good first choices for him are Strength and, in time, Move speed.

The different buttons at the top row cycle the different sets of items in the game. Consumables are generally your support items. Next to that are stat-boosting items. The sword and shield represents Damage / Armor boosts, and the final button is Arcane items that boost your other stats (Evasion, Regeneration, etc). Left clicking any given item will show it, and any "recipe" items involved with it. For example, under the Starting Items, you see two Gauntlets of Strength. One of these can be combined with a Circlet to create the Bracer you see under "Early Game". You can right-click the Bracer and let the game auto-buy the ingredients to be auto-combined, or you can purchase the different items over time yourself based on what you need when.

One important note that will be covered later is the common "Requires Secret Shop" marker that shows one of the ingredients will be bought only at one of the two Secret Shops on the map. This is covered later.

When you find your lane and battle starts, expect a few creatures called Creeps to head out into their respective Lanes to do battle. You must be next to these fights (even if not participating) to get Experience from it. However, you must deal the last blow to any enemy to get the Gold from its kill. Take note, Ranged heroes have it safer than Melee heroes because they are less prone to tanking damage from Creeps. A Melee hero rushing creeps for gold is open to a Ranged spell that can severely cripple the Melee hero's early game training.

Check the Minimap in the bottom left to see where the lanes are respectively. X marks the common starting battle locations where you can expect Creeps to meet, and Heroes to first appear. The most common setup is two Heroes to Top, one (almost exlusively Ranged) to Middle, and two to Bottom. If a Jungler hero wants to go into the Jungle early, their lane may be a "suicide lane" (unofficial term) where the Hero defending the lane must be ready to fight hard or flee, and where the Jungler must be ready to join a fight on a whim to save the lane.

Where fighting is concerned, two notes to keep in mind. Don't tank--Creeps to a startlingly high amount of damage to non defensive Heroes. And don't get too close to enemy Heroes, since the most common first picks for Spells are those that allow enemy Heroes to get an edge in a fight against you. Killing a Creep is not worth being stunned, slowed, and ganked (killed) while still level 1.

When one side gets an advantage, they start to approach one of the most important structures in the game. The Tower.

This structure has a large range of fire, can see invisible targets, is immune to Spell damage, and never misses. If you absolutely have to fight, fighting with one at your back is a good thing. Towers are almost impossible to tank alone. Pushers often have minions or clones that can take the damage, and pushing with many Creeps on your side is the only safe way otherwise. Until you can sneak in and deal high damage to it consistently, expect to follow a swarm of ~3 Creeps in, attack about four times, and run away.

Towers are particularly dangerous to move past. The front most tower in a line is vulnerable to damage. Any structure on the Lane behind it is invulnerable. If you manage to get behind enemy lines, you still can not destroy any important structures until the lane is worked at from the front. Towers are the first step. Destroying one gives gold to all your allies (and Denying one prevents this boost from going to your enemies).

Last Hitting an enemy will visibly show how much gold you gain in a small display. Use this marker to watch your success with hitting at the right times, because Creeps like to steal these kills.

A useful trick if you have the time is to left-click your enemy Heroes to scout some information about them. Their total stats, inventory, health, and mana are able to be read while you have them selected. This especially comes in handy with any Heroes that have items that are dropped on death, though any information is still useful. After all, if you know the enemy can't heal anymore, they're more sure to retreat if threatened.

The "Channeling" feature is specific to heroes who have Spells that deal with it, and it works differently for each Spell. One feature always remains the same. If you move, at all, the Channeling will end and the perk of the Spell will be lost until you cast the Spell again and let the timer finish.

Sand King's example is a sandstorm. As long as the Sandstorm is happening and Sand King stands still, he is invisible in the center of it and deals AoE damage once per second for the Sandstorm's range. Other examples of this feature include Tinker removing his cooldowns after a very short Channeling sequence, and Lion's Mana Drain Spell. So if any of your Spells describe being Channeled, expect to have to stand still for a short time.

Heroes die. It happens. In this case, pursuing a weakened enemy Hero behind enemy lines to try to kill ended in my own Hero dying to a blast from the Tower. Because the kill was made almost exclusively by the Tower, Earthshaker was not given a benefit for the kill. The camera can still be moved while you're dead, and you will respawn in the Base, so the Shop will immediately be available to get items from. However, the camera tends to replace itself back on the location you died, so expect to move it once you've respawned.

At the top and bottom, you can see a timer that shows how long you have until respawning. Buyback will immediately buy you back into the game, though this comes at a usually high cost and with a cooldown until you can Buyback again. Most players instead spend their gold buying items during death and claiming them upon respawn.

Your respawn times get higher as you level up and as you die more.

In this case, I didn't buy anything for my character because I had to return immediately to the Middle tower to defend it. I did, however, buy an Animal Courier (a little creature that delivers items to one Hero at a time) in case I needed anything while gone. As it turns out, I experienced a little trouble by the tower that sapped half my health because I was unable to run away from the Tower quickly enough. So my decision was to buy Boots of Speed while out in the field. The problem is that buying in the field keeps you from getting the items until you're back at the shop. So, with items bought in the field will be kept in the Stash until you return... or until you use the Courier. The little box at the bottom right showing an icon of a donkey's face is the courier select button, and the one below it commands it to scoot over and give you all the items in your stash. So click it and becomes to show delivery to the hero shown in the photo. A few seconds later, the Stashed Boost of Speed were delivered, and the Courier was skittering its way back to base.

Do not be afraid to buy items outside of your Base. Dying unexpectedly may cost you the Gold you need for a good item, so keeping your Stash full and wallet empty to rely on the courier later is not a bad idea.

By the time you may get to exploring Courier use, you'll probably wonder and start to learn when a good time is to rush a Tower. This is the result of a good scenario turning bad. No enemies, no Hero (often called "Missing ___" where the lane is filled in), and what started as three melee creeps, one ranged, and one catapult was dwindling rapidly. Most of the damage to the Tower was dealt during that one rush, and escaping as the last Melee creep is a good idea because you will otherwise become the next target of its attacks.

If it hasn't been made clear by this far into playing a game, Creeps are meat shields. Take advantage of them as greatly as you can.

As shown with small pink circles on the map, the two Secret Shops are fairly accessible by either side. A Dire player is not limited to Dire shops, and vice versa. (Fun fact, if you manage to get to your enemy's base and hover just outside their anti-spawn camping range, you can still use their shop like normal.)

Secret Shop items are unique and almost always involve pure stat boosting or regneration. The items in a secret shop will almost always be involved in making the game's strongest items, and can not be delivered by Courier or bought if your inventory is full. You have to have an empty slot in order to buy its items, so in advance, prepare to go to the Secret shop first to directly buy items required for your best recipes.

The other location based perk will lie in where Runes spawn. Runes appear where little pink circles are marked on the Minimap and come in five useful varieties, once every two minutes and never the same rune twice in a row.

Double Damage - The Hero (and anything under the Hero's control within 500 units of space) gets a 45 second, 100% boost to normal attack damage.
Haste - The Hero gets a 100% Move Speed boost for 30 seconds. Haste runes prevent Move Speed slowing.
Illusion - Two illusion clones of the Hero appear for 75 seconds or until destroyed, dealing 35% of the Hero's damage and taking 300% what the Hero would.
Invisibility - The Hero goes invisible for 45 seconds or until the Hero attacks (including using a Spell). True Sight will see through (but not nullify) an Invisibility boost.
Regeneration - Until the hero takes damage, (s)he gets 100 HP per second and 67 Mana per second for 30 seconds, enough to almost assure full recovery.

The Bottle item can store Runes. Using the Bottle will manually apply the Rune, and they will otherwise be automatically activated in two minutes as a new one spawns. It is common for a faction to place Sentry Wards in a manner that lets them view the Rune spawn points.

After much struggle, killing enemy Heroes becomes a possible option that should be taken advantage of. A Hero dealing death to another Hero is very important to the victor because the experience is immense and the gold flow will help shape the game. The victor will almost assuredly level up (to the point that they might level twice, if making the kill while already near a natural level-up), and will gain much gold while causing the dead hero to lose some in the process. This is why deaths are to be avoided, caution is suggested, and Heroes that can gank enemies easily or unexpectedly are particularly dangerous.

To continually die, whether on accident or purpose, is called "feeding" the enemy. A single hero may become level 25 (the maximum) well in advance because they gain an early advantage due to having killed their opponent quickly enough. The strong indeed get stronger if you manage to be on the winning end of fights.

After enough back and forth encounters, you'll finally have destroyed enough Towers to make it into the enemy's home camp. Behind the final Lane are two Barracks. One for Ranged creeps, one for Melee creeps. Destroying one on the enemy faction's side allows your creeps of that lane to become boosted.

Example: Destroy the Ranged barracks of your enemy's bottom Lane, and your bottom Lane's ranged creeps will become "Mega" ranged creeps. Doing this to both Barracks will almost assuredly allow that Lane to become a permanent problem for your enemy, as a swarm of Mega creeps can and will tear through the enemy's camp if left unchecked.

The stats for creeps are as follows.

Melee creep - 21 Attack, 2 Armor, 325 Move Speed
Mega melee creep - 46 Attack, 3 Armor, 325 Move Speed
Ranged creep - 29 Attack, 0 Armor, 325 Move Speed
Mega ranged creep - 52 Attack, 1 Armor, 325 Move Speed

Once you finally go through the Towers of any given lane, the two Towers defending the enemy's Ancient will be vulnerable. The game calls out the Ancient's defensive Towers as "Middle", but they can be exposed by clearing the Towers of any given lane. Once the two Towers are destroyed, the Ancient is vulnerable to damage. By this point, the game is almost assuredly won by whichever faction has exposed the Ancient. Pulling an upset is possible, but highly unlikely. Also, by this point, the only defence for the Ancient will likely be your faction's Heroes.

After a lot of effort, averaging 45 minutes in a proper game (and up to an hour in a constant struggle for an advantage), one side gets victory and the results are shown. All the items of all sides, the kill counts, and... well, every bit of possible information about the game except what specific Spells were trained and who killed who, when.

An online game against people will potentially result in Items (cosmetic) being randomly won by a player. Battle Points are also earned, and if the 1,000 point / level-up mark is reached, the player is assured an item drop for that game. Aside from that, they have no visible use thus far except potentially helping a player guess how many games friends have played.

Disclaimer: Actual mileage may vary. Passive bots, as used in the demonstration, do not use Spells and rarely push an attack. Enemy players come in many varying degrees of skill and any level higher of Bots will be able to converge to push against your faction.