I'll try to give you a condensed version.
Mafia is a forum game where there are two faction.
The peaceful and majority town and the villainous mafia. Each user will be given an alignment and a specific role/power. Town users want to erase any threats to the town and need to try and vote to lynch potential town members. Meanwhile, mafia members want to outnumber the town and kill them during the night and mislead lynchings.
It's a game of mind games and deception. You need to determine who you can trust and who you should avoid fast.
There are two phases in the game. Day phase and night phase.
During the day, users can speak publicly in the thread (and only in the thread) and discuss ideas, strategies and claims. At the end of the day, the town will lynch someone, so townies need to vote on who they lynch.
During the night, users cannot speak in the thread, but can PM each other to discuss things. During the night, the mafia decide who they want to kill that night. At the end of the night, if all goes well, the mafia kills that user leaving the town to find them dead in the morning. This is also the time when most players can use their powers. These powers can range from anything from checking alignment, preventing a kill, preventing someone else from using their power or checking bits of other people's PMs.
This continues until the mafia is eliminated giving the town the win, or until the amount of mafia members equal the town members giving the mafia the win.
To add to that, many games have third parties, who have their own separate goal to complete.
You might have heard of the game before as a party game or several online versions such as Epicmafia or Town of Salem. At times it has been called Werewolf.
You can learn more here:
https://wiki.mafiascum.net/index.php?title=Main_Page
It's got lots of specific roles and more detailed explanations.
It might seem seem like a lot, but it is super fun.
Back when we had a Smash 4 Community Hangout, we played it almost constantly. It was incredibly fun and deepened a lot of relationships I had with other users.
Plus we even managed to make themed games too. We did Ace Attorney, Professor Layton, FNAF, Godzilla, Team Fortress 2, Pokemon, Smash Bros, Dangaropa, etc. Introduced me to a couple of new series while I was at it.
It'd be great to play it again.