Short Explanation:
It is a game where you call people idiots, assholes, and liars (in good fun).

Long Explanation:
Mafia is a party game modeling a battle between an informed minority ("the Mafia" or "scum") and an uninformed majority ("the Town"). Member of the mafia know who each other are, and usually are able to communicate with each other via private methods. Townies do not know who each other are but they greatly outnumber the mafia. The goal of both factions is to eliminate each other. More complicated games can have more than two factions, as well as factions with victory conditions that are different from the standard "kill everyone else."

Before the game starts, the moderator will send everyone their role in private. A person's role consists of which faction they are part of, as well as any special abilities they may have.

The game consists of two phases: Day and Night.

During the day, the players must discuss and collectively choose someone to lynch by voting for whomever they think needs to be eliminated from the game. Townies will try to vote off people whom they believe to be Mafia members, while the Mafia will try to lead them astray. Once the majority of people have voted for a single player, that player will be eliminated from the game, and their role and alignment is (usually) revealed to all players.

During the night, all public communication stops, and the Mafia collectively chooses someone to kill, and submits their choice to the moderator. In addition, any players with special night actions submit their decisions to the moderator. Common night actions include investigating other players to find out their alignment and protecting other players to prevent them from dying that night. After the moderator has received all night actions, the moderator publicly reveals who died during the night, and a new day begins.

The game continues to alternate between day and night until one faction has achieved its victory conditions (usually eliminating all other factions)