Dota 2 is a MOBA, or multiplayer online battle arena, game. The short definition of a MOBA is a game in which two teams face off on a map with the goal of destroying the opposing team's base. Most of them include non-player creatures (creeps) that move from each side's base to attack the opposing side, and the destruction of said creeps is a primary source of gold and experience for leveling up the player controlled heroes. League of Legends is currently the most famous example of the genre, but the original Defense of the Ancients mod of Warcraft 3 is what got everything started.
If you've never played before, Dota 2 now offers a fairly solid training course in the game itself. It will familiarize you with the controls and some of the mechanics in a stress-free environment. Play the training levels, it'll help. I wish they'd been there when I was starting.
I'm going to assume you understand the basics of RPG-like games, of gaining xp, leveling up, earning gold and buying items, etc, and that you know how to start the game, jump into a bot match, and pick a character. With that in mind, I think the single most important thing you need to learn when starting out is the concept of "last hitting".
Every time an enemy creep dies, if you are close enough to it your hero will gain experience. However, gold is only awarded to the hero that lands the death blow on the creep, or the "last hit". Doing so also gives more experience than just being nearby. So if you get lots of last hits, you will make lots of gold and get lots of experience. If you don't get any last hits, you will make small amounts of experience and no gold at all. Last hits are key.
What you absolutely do not want to do is wade into a pack of enemy creeps and auto attack until they're all dead. If you're by yourself, you will take a lot of damage. If you have the support of allied creeps, you'll get very few last hits. In the early game, timing is everything. What you actually want to do is follow your allied creeps, allow them to engage the enemy creeps, and when an enemy creep has just a sliver of health left, attack it and get the last hit. If you attack too early and don't kill the creep your creeps will finish it off. If you attack too late, the enemy creep will already be dead. Timing is very important.
The opposing heroes will be doing the same thing. Rather than just stand around waiting for last hits, you can try to harass the enemy heroes with your abilities. Keeping them low on health will force them away from the creep line, and starve them of vital experience and gold. Note, however, that using your auto-attack (right click) on an enemy hero will cause nearby enemy creeps to switch targets and attack you. Harassing the enemy is important, but you have to be careful.
Another important thing to keep in mind is hero selection -- some heroes are much harder to play than others. Personally I think ranged heroes are easier when starting out than melee heroes, as you have more time to respond to what the enemy is doing, and tend to stay out of immediate danger. It's also important to know the role of the hero you choose -- many heroes are "carries", designed to dominate the enemy team in the end game, but typically quite weak in the early game. Each team only needs 1 or 2 carries at most. A team with 5 carries is probably in trouble. The Library tab inside Dota 2 defines the roles for each hero. If you're playing by yourself, a carry is fine, the bots will do their best to support you. If you're playing with others, see what they've picked before you pick.
The other key thing to be aware of is building items. When I first started out I didn't understand that every big item in the game is built from a number of cheaper items. You don't need to save up 5,050 gold for that Bloodstone, you just need to buy each piece that constitutes a Bloodstone, and once you have them all, they transform into a Bloodstone. Never ever ever "save" money for a big item -- you should be buying or upgrading items every chance you get, from small things in the early game, to expensive things from the Secret Shop in the mid to late game. The more items you have, the faster you can earn gold, and the faster you'll get that big item you really want.
For a detailed schooling, you should read this: Welcome to Dota, You suck. It has almost everything you could ever need to know about how to play, in great detail.
Another great link is the Dota 2 wiki, which provides overviews of all the heroes and good tactics to use with them. It also provides detailed explanations of some of the mechanics, such as the difference between Pure damage and Physical damage.
Learn the game, practice, and before you know it, this could be you!
No, no, I'm kidding. Don't be that guy, Witch Doctor is only that good with careful video editing . . .