What are the different leagues? I haven't spent much time looking into anything other than just playing.
There is standard, which is just the normal game, hardcore, where it's only characters that haven't died. Unlike other games like diablo though, if you die in hardcore you go to standard, your character doesn't get locked.
There are currently also 2 4 month leagues:
Domination, which is the default for new players. It's like standard but with the added buff shrines throughout the world and unique items that you can only get in domination.
Nemesis, which is hardcore but also has nemesis specific monster "mods" to make it even harder.
They, at least in beta and I assume they will continue, also periodically add leagues. For instance they had cut-throat leagues before. In cut-throat, instead of you having your own instance of zones, they were all public. Meaning other players could freely join no matter what (you couldn't be private if you wanted to). In addition, full pvp was turned on, so you could get killed. Add on top of that, on death you dropped all your items. So basically you could be assaulted and robbed of all your equipped gear (or assault other players for their gear)
They also have races periodically, where it's only a few days or hours even. A lot of times, in beta, there were prizes for completing challenges or being the top players in a race, such as signed posters or t-shirts and such.