The Ultimate Guide to the Chatterbox calls an RP "A story in which multiple CBers participate. Each participant has a character which they create based on the topic and then everyone works together and writes a comunal story from the prespective of their respective character. Usually written in first person, but it's not a requirement." 

So typically, someone will make a post with the beginning of a story, and then people wil join by commenting their filled-out character forms. Then everyone takes turns writing a piece of the story from their character's perspective. The person who started the RP ususally has some control over where the story goes, but it's made up as you go along. Try reading a few posts with RP in the title to get a feel for it. I hope that helps! 

You can find the Ultimate Guide here: http://www.cricketmagkids.com/chatterbox/chirpatcricket/node/204967