Sure thing. There is mafia, which is a small portion of the players. They know who each other are, and want to kill all the players. They can attempt to do so with Night Actions, which happen during the Night Phase. More on that later.
They can also attempt to Lynch a player, which happens during the Day Phase. Lynching is when a person is voted to be killed. As in: 'I think that guy is a mafia guy, so I vote That Guy'. The Town (the portion that is not Mafia) wants to kill the mafia, by Night Actions or Lynching.
There are two Phases, Day Phase and Night Phase. The game usually starts in the Day Phase, during which players choose who they want to lynch by voting for someone. Then the Night Phase happens, where Town and Mafia send in their actions.
Different Town and Mafia players have different roles, and different abilities.
For example, a person with the role of Cop who is on the town side can investigate a player, and find out if they are town or mafia.
A person with the role of Surgeon can save a player from death by the mafia or another player that night.
The mafia get to choose who to kill during night phase, similar to lynching except they don't post in the main thread. Instead, they PM each other and the GM on who they want to kill.
The roles people get are usually private and randomised. Someone might be a mafia, they might be a cop, they might not have a role (This is usually referred to as being a townie).
Most of the game is spent analysing other peoples posts. If someone is acting oddly, maybe they are trying to conceal their status of being mafia, or maybe they are just odd. Then, people will lynch each other based on their posts, attempting to find the mafia. In a Role heavy game, where most of the players have a role and are not townies, sometimes people will prefer to wait for the Roles to find out who is mafia and not, and then lynch based on solid evidence.
That's about as simple as I can make it.