The origins of League of Legends can be traced back all the way to a Blizzard game, Warcraft III. Part of this game (which was actually a real time strategy experience) was the ability to edit maps and from this players would create their own custom scenarios or maps for other players to enjoy.
One of these custom maps was so in depth in design that it actually changed the game objectives completely and this was Defense of the Ancients or DotA. While the game grew rapidly and changed hands several time it was eventually let down by the fact that it was a game within a game and lacked the extra features that it needed to really see mainstream success.
The game is still played today (and I still love it) but the restricting features meant that the game never achieved it's potential in my opinion. It's from this idea though that League of Legends was born with participation from one of the administrators that managed DotA for an amount of time.
Now that League of Legends has gone big a large number of different genre titles have been applied to the game and the genre as a whole.