It really depends on the type of game and the GM (person who hosts the game), most of the games have a 'sign-up' thread that explains the rules and the way to play. I think that the 'forum game' section of the forums is currently being a bit flooded with Commander threads, which is a kind of card/strategy game (Clicky here for a single player game which also describes the rules, I think).
Other popular games include 'mafia' games (which are my favorites) in which players have to uncover the criminals amongst them by asking each other questions, talking to each other in the thread, etc. while the 'mafia' try to kill them all.
Most of the forum games on this site are so-called RP's. These are role-playing games in which players have to create their own characters, including back-stories and abilities. These kind of games are usually played by players telling the GM what their character does and the GM telling the players the consequences of those actions. So you could kinda see it like a forum CYOA, only without a back button.
Of course there's also more 'simple' game threads like riddle threads in which people ask each other riddles. These usually don't have a sign-up thread and anyone can but in at any time.
I know this list is far from complete, so I'd advise you, if you really want to know how to play forum games, to join a few of them :)