Basic Rules
In its simplest form, Mafia consists of two sides: the Mafia, and the Town. The Mafia's goal is to kill the innocents, while the Town's goal is the kill the Mafia.
At the beginning of the game, the Moderator will secretly send everyone their role. The Mafia will know each other, while (in the basic game) everyone else will only know their own role.
The game has two phases. Usually the first is Night. During Night, the Town can not communicate, but the Mafia can discuss and choose someone to murder. After the Moderator has received the name of the victim, the Moderator reveals the dead player(s), and Day begins.
During Day, the Town must choose someone to Lynch; they are trying to eliminate the Mafia, but the Mafia can lead them astray by casting suspicion elsewhere. Generally, players will vote for someone they think is Mafia (the Mafia will vote so that they seem to be doing this as well, and might even vote for each other to confuse the Town); when a player gets a majority of the votes, they are lynched. Their role is/isn't revealed by the Moderator, and it is once again Night.
The game alternates between Night and Day until either the Mafia are eliminated, or the Mafia have killed everyone else.
Basic Roles
The above suffices for a Vanilla game, but most games have roles other than just Mafia and Townie. These roles have additional abilities (which are usually used during the Night, though there are some Day abilities as well). The most common are listed here.
Cop - Each night, the Cop can investigate one player (by sending their name to the Moderator), and finds out whether that player is Mafia or not. The Cop is on the Town's side.
Doctor - Each night, the Doctor can choose one player to protect (by sending their name to the Moderator). If the Mafia (or anyone else) tries to kill that player, they will fail. The Doctor is on the Town's side.
Vigilante - Each night, the Vigilante may kill another player (this varies) (by sending their name to the Moderator). The Vigilante is on the Town's side.
Mason - The Masons are a group on the Town's side who know each other and can talk to each other at Night.
Serial Killer - The Serial Killer (often abbreviated as SK), like the Vigilante, may kill another player each Night (by sending their name to the Moderator). However, the Serial Killer is not on the Town's side or the Mafia's side. The Serial Killer wins when everyone else is dead.
Multiple Families - Sometimes there is more than one group of Mafia. These act independently of one another, and must kill the other group as well as the Town in order to win.
There are many variations on the above roles that will often appear in more complicated games.
