It's basically an advanced Guess Who game, here's how it works.
There are two teams: Town (good guys) and Mafia (bad guys)
The Mafia players all know who their teammates are but there are only a few of them. The Town players only know that they are Town and don't know what anyone else is but there are a lot more of them than Mafia players. The goal of each team is to essentially eliminate the other team.
The game is played in two phases: Day and Night
During the Day players chat in a game thread and decide together who they think is Mafia. They do this by voting to kill a player (called lynching in the game) and when a player has the majority of votes the game moderator pauses the game and reveals whether that player was Town or Mafia. That player is now dead and out of the game and the game then proceeds to the Night phase.
During the Night the Mafia discuss privately which Town player they want to kill. When they've decided they submit their kill to the game moderator who then announces that the player is Town and has been killed by the Mafia. Game then goes back to Day phase and the process repeats.
The game continues until either all the Mafia have been lynched or enough Town players have been lynched or killed that they can no longer reach a majority decision to lynch a Mafia player, at which point the moderator would call the game a Mafia victory.