Mafia (also known as Wolf) is a game of logic. There are traditionally two sides: pro-townies (the good guys) and a smaller group of anti-townies (the bad guys). For the examples in this post, the pro-townies are humans and the anti-townies are demons.
The game is run by a game moderator. Each player knows his/her own role, but not the roles of the other players at the beginning of the game.
The game has two phases: Day and Night. During the Day phase, players discuss in the game thread and try to figure out who is a demon. Each player gets one vote and the player who receives a majority of the votes will be killed (eliminated from the game). Then the game moves to the Night phase. The demons find out who each other are and must choose one player to kill during the Night phase. They PM the mod who they chose, that player is notified that he/she was eliminated, and then Day 2 starts.
The humans win when they have eliminated all the demons. The demons win when they have eliminated enough humans that there are equal numbers (2 demons and 2 humans, for example) and they control half the votes.
Here are a few sites so you can familiarize yourself with the game. We may run games differently, but these sites will give you a good idea of how the game is played.
History of the game
Newbie Guide at the xkcd forums
Newbie Guide at Mafiascum
If you have any questions, please feel free to send a PM to Angel325girl or Seraphim.