Basically, I don't know if you've ever played an online text-based Role-play, but it works in a similar way. You sign up with an account, and then create a character you'd like to have (For me, I have Parker for example, who is part student/part super-spy) and then you just write what they do, say, etc.
So it works like this:
player 1 said:
Graham: Hi! I'm new to this neighbourhood...
player 2 said:
Brian: ...er...hi there...I've lived here for years, but am too shy to get involved with the community...
player 1 said:
Graham: Well, let's check out the local coffee-house together.
player 2 said:
Brian: *walking away* I do declare this may be the most boring conversation I've ever had...
How it works on our site specifically is that there is a Topic on the forum where the story happens, and you just interact by saying what characters say, and doing what characters do.