Mafia is, at its simplist, a game between an uninformed majority and an informed minority.
At the start of the game, the moderator divides the players into two teams, the townsfolk (who do not know who their teammates are), and the mafia (who do know who their teammates are). The mafia:townsfolk ratio is generally between !:3 and 1:4.
The game is divided into two phases, day and night. Each day, all the players discuss who they believe to be in the mafia, and vote on someone to lynch. Once a player attracts the majority of the votes, they are killed, and their alignment is revealed to the rest of the players. At night, the townsfolk are not allowed to talk, and the mafia privately discuss who they want to kill. In the morning, that person is announced to be dead, and their alignment is revealed to the rest of the players. Dead players cannot talk or vote.
Rinse and repeat. The mafia win when every town player is dead, or when nothing can stop this from happening. The town win when all the mafia are dead (there is never an 'or nothing can stop this from happening' for the town in standard mafia).
For those of you who don't like reading: http://www.sitesled.com/members/mikeburnfire/mafiascum04.swf