The best way to learn Mafia is to play,
but here is a basic explaination to help.
Mafia is a social deduction party game that presents a conflict between two teams: The Mafia (an informed minority) and The Citizens (an UNinformed majority). At the start of the game, each player is secretly assigned a role affiliated with one of these teams.
The game has three alternating phases: Covert Operations, Open Floor, and Voting which are repeated in sequence until the game is over. During Covert Operation (Sleep Time) The Mafia covertly kills one citizen. During Open Floor all surviving citizens debate to determine which players they believe are secret members of The Mafia. After debating, all players vote with the intent to eliminate a mafia member. If the citizens wrongfully accuse and vote a citizen off, the citizens loose that team member. The game continues until one of the teams achieve their objective: elimination of the other team.