This area is located in New York in the game. When players arrive here, an option on the side bar will appear saying "find a fight". The game then will match the player with another player around the same level who is also looking for a fight. The fight club is the only player-to-player fighting feature in the game.
Players in a Fight Club fight have two options: change target or forfeit. Players can't actually "run" from another player after initiating a fight and they'll be followed around unless they change their target, forfeits, or someone is KOed.