Leage of Legends is a 5v5 Arena game. Dota (Defence of the ancients) is it's mother-game (mother of most AOS games). Another similar game is Heroes of Newerth (more in-depth than LoL and better graphics).
Your mission is to work with your team and defeat the opposite team. You select your character (a list of heroes) and begin the game. There are several shops where you can buy items to increase speed, get new abilities and much more. You level by killing mobs (auto-respawns from enemies lair) or by killing heroes (other players, gives more experience). Whenever you level, you can increase your skill levels by 1. If you are killed, you will respawn in your base after X-time (different in all games, I don't know what LoL's re-spawn time is but I guess it's based on your level. High level = slow resapwn, low = faster).
Main mission: Destroy the other team's base, working with your team to success.
Search YouTube for tutorials.