Roleplay (or role-play) is what happens when several people get together, and each takes on the role of a character (original or canon), whose actions they narrate. This can be a lot of fun if done well, or it can descend into Sueishness and poor characterization. Sometimes roleplay chat logs are written up into RP fics. There are several different kinds of roleplay.
Roleplay on the Internet is usually a text-based roleplay on a forum or in a chat room. These usually have few rules and rely primarily on the honor system on the part of the players so that their characters don't become so god-like that nothing bad can ever happen to them or, worse, that they write things happening to other people's characters without their consent. This sort of behavior is called power-playing or godmoding, and is heavily frowned-upon. Perhaps even more frowned-upon than creating a Mary Sue, which is also really annoying. If there are mods involved, godmoding is likely to get the guilty player kicked out of the RP.
Online RPs can generally be split into interactive online stories and chatplays.
Interactive Online Story
This type of roleplay happens on a forum and usually not in real time, which allows for documentation of the canon/original setting and long, narrative-style posts with attention to details. Character information will also be recorded somewhere as a reference for the player and anyone else with an inquiring mind. Sometimes an IOS will have one person in charge of the overall plot, or it will have a group of mods who work together to ensure that there is plenty of activity and the rules are followed. In the former, there tends to be one major storyline with perhaps a few sub-plots; in the latter, there are many smaller plots running between various small groups of players, all of which contribute to the overall environment of the game.
Occasionally, there is no one person in charge, and all the players work together cooperatively to create an interesting plot for their characters, much like co-writing a story. Usually this only works with smaller groups of people who get along well, though.
Chatplays take place in real time over a chat or instant message program, or even a forum if enough players happen to be on it at the same time. The defining characteristic is that it happens in real time, or close to it, and there is no permanent record of it unless someone takes the time to make a transcript. Since the pace of the action is much quicker, chatplays lend themselves to silliness and crack, and in instant messages or chat clients especially due to character limits, they often take place in script format with an emphasis on dialogue and actions noted briefly in *asterisks*. However, with a good group of people they can be just as deep as other types of RP.
It is still not usually a good idea to publish of these up as RP fic without first giving it some serious thought and attention.
Because the website/game ROBLOX utilizes a scripting language called Lua, many ROBLOXians have created morphs, which are models that have a pad in front of it that, when touched, turns the player into the model. These are usually used in roleplay places.
An easy type of morphs are Paper morphs, with the models simply being flat bricks with a picture of whatever is meant to be of on both sides.