This is what the game gets its name from; obviously, it's the most important role and is in almost every game. The Mafia usually know each other and can talk to each other at night. Each night, they choose a victim from the rest of the players. When day breaks, the victim is dead. The goal for the Mafia is to survive while killing everyone else in town.
A very common variation on this role is to have two (or more) Mafia families, sometimes called factions. Each family operates separately, and are not on the same side; a family only wins when everyone else, including the other family, is dead.