The short of it:
Prior to the game starting, everyone is assigned a role. The majority of the roles are "innocents", some of which have added special powers which will be known to everyone. There will also be a minority of players assigned to the "mafia", who have the power to kill one non-mafia player every night.
Every day, all players in the game select one person to vote for, and the person with the most votes at the end of each day is killed off. Obviously innocents want to find and kill the mafia, and the mafia wants to preserve their numbers.
The innocents win the game if there are no mafia members left alive. The mafia wins the game if they ever outnumber the innocents.
In the past, one of the recommended ways to see how things work is to read a past game thread or two. Here are three that are different enough from each other if you wanna go back and study a bit:
Calvin & Hobbes Mafia (the last one I mayored)
Price is Right Mafia
Ace Ventura Mafia