Mafia is a game of deceit. It consists of an uninformed majority vs an informed minority. In other words, town vs mafia(sometimes, you might hear it called village vs wolves too). The townies are the good guys and the mafia are the bad guys. The game is split into two phases: day and night. The day phase is where everyone can talk in. Players can vote to lynch another player. At the end of the day or when majority is reached(either or, depends on the game), the player with the most votes is lynched and therefore, dead. This phase is predominantly for town to try to eliminate mafia. The night phase is where players turn in their actions, this is mostly used for mafia to kill townies. Town wins when they have eliminated all the mafia and mafia wins when they have outnumbered town(or sometimes equaled them, whichever comes first).
There are many more variables in the game depending on the complexity of the setup, but that is all you need to know about mafia for when you start. You will learn more as you play the game. If you have any questions, do not hesitate to ask any of the other players or the host of the game.