Short version, it's two or more people writing a story together.
Here's the long version:
RP stands for Role Play. It's a type of writing which usually happens between two or more people, and it is notoriously hard to finish a single story before starting a new one. Each person takes on one central character as their character, and then accepts responsibilities for several or many others, depending on how many people are participating in the RP. However, the more people, the harder it is to keep organized and controlled. Traditionally best and easiest between two people, usually taken on as a way of bonding between two writers. One person starts, the other replies, the first replies to the reply, etc. Often taken on in the writing community between two people who are close friends, or else is used as a way for writers to get used to another person's style. When the two people participating in the RP know one-another very well, it's often possible to distinguish what their reply to a certain situation will be, and can therefore be anticipated and detailed before it even happens.
Manners in an RP are actually fairly complicated. Here are the top five main rules that pretty much everyone obeys when it comes down to it...
The first and most basic rule is that another person's characters are practically sacred. You are issued your own characters, and you are allowed to get indignant if someone else tries to mess with them. Be sure to ALWAYS check with your partner before fricking around with one of their minor characters, and NEVER try to somehow control another person's main character unless under very extenuating circumstances. Many people have found themselves banned from an RP for a single instance of it.
Second rule is, when someone asks you if you want to continue the RP, pay close attention to your mood. If it's a serious story and you're in a playful mood, say you can't do it. Don't even try doing it in the wrong situations, because there are situations where someone will get furious with their RP partner for messing around. This also includes the fact that you should try and keep in mind the laws of physics in whatever world you're writing about. No magical, spontaneous resurrections of characters in worlds that don't have magic unless it can be explained with science. This rule really should never be ignored.
Third rule. Don't do too long or short of a reply. You may feel the pressing need to add more and more detail, but it makes the other person wait, and their character doesn't get to interact at all during that time. Also, if it's too short- come on! The other person has to reply! Give them something to reply to! This also includes the fact that every bit should try to have at least a little speech on their part, or in cases of a mute or very quiet character, at least an action the other person can reply to. If nothing else, just have them curl up in a corner and show their thought process. This rule can be ignored under certain circumstances, the biggest one being when it's at the beginning of a story and both people are attempting to establish their characters before they get to the meat of things. Another situation is if the characters are separated temporarily.
Fourth rule... please. As grammatically correct as possible. No l33t speak, no ignoring punctuation, and always make sure things are spelled right. It just makes you look like an idiot, and the other person will end up silently frustrated with you. This also includes not naming your character something long, ridiculous, and is unable to be pronounced by any mortal tongue. Some immature RPers ignore this rule entirely, but it makes everyone else roll their eyes.
Fifth rule. This one can be ignored, but most people at least offer it. It has to do with noobs to RPing. If one person has done an RP before, and the other hasn't, then the more advanced one often suggests that maybe they should take on most of the characters, leaving the noob with one or two to play around with on their first try. This can be ignored under many circumstances- such as if both are new, or one is new and the other has only done a few RPs. It's mostly advanced RPers who offer it.
One last thing to note is that people often have a symbol to show the breaking of the fourth wall, such as *. Try to make it as smooth a transition as possible... don't refer to it as 'my character' or 'your character'. If you want to suggest something to the other person, use *. Such as like this: You want to offer that maybe their character should get captured. You reply, and then at the beginning or end of your reply, you put in *Hey, what if insert name here gets kidnapped?*. If they don't go for it, don't push it. In that way, you can trust that the other person won't push their unliked ideas too hard either.
A little tip. Your main character is often referred to as your Chari. It's just a piece of slang that's developed.
Person B is an example of what NOT to do in an RP :
Person A: James walked through the door, looking around nervously. He had to find Sharlene and get out of there before anything else popped out at him. If he ran into another one of those ghouls (he gulped nervously at the mere thought of it), he didn't know if he'd be able to make it out in one piece...
Person B: LOL ha ha a ghoul comes out and stabs James in the stomack.
Person A: No, it doesn't. As the ghoul comes toward him, James backs off until his back was pressed against the wall. When it lunged at him, he jumped aside, heart beating wildly as he reached out. His hand caught on something, and he raised it high without thinking, bringing it down on the ghoul's head. It fell to the ground, twitched once, and lay still.
Person B: Then the ghoul gets up and says "HA HA HA! I don't need a head, bitch! FEAR ME!" and he transforms into my chari Damien Von Glukthenhiemerwhatsisname the 3rd and stabs your chari and James dies of blood loss while Sharlene gets butt-raped by zombies.
Person A: *...*
Person B: What? Why didn't you reply?
Person A: *... I'm going to go have dinner and I'm never coming back.*