Role playing is, as the name suggests, is playing a role. You say you are a character from an anime, or even from some other fandom, and act out that character. For instance, one of my absolute favorite characters is Arisawa Tatsuki from Bleach, and on Facebook. I role play (rp) her.
There are several different ways to rp, but there are two main ones. First one is to do the single sentence, action or word answers. For example
Tatsuki: Hi Ichigo, how are you? *she waves at him as he walks towards her*
Ichigo: I'm doing fine and yourself? *he swung his bag over his shoulder and he raised his other hand in answer*
Tatsuki: Doing great thanks.
Ok, I know that's a lame conversation, but I was just making a point. There was just a single action and sentence spoken in each reply. The other kind is when you write a full paragraph with several actions and sentances in each.
This one was an rp starter that I actually had responded to once a long time ago.
Those kind of rps can be a lot harder, but are usually a lot more interesting. For those who are not used to writing can have a hard time writing out a scenario like that. But it is so much fun to work around each person and include their comments and actions into your response.
Now, a possible problem with rping, is that sometimes people don't act like their character and all they're after is to hook up some random guy or girl. They take the name and picture of a character, and then don't act like them at all, just themselves. At least that's the case on Facebook. There, nobody knows the meaning of true rp and so it isn't nearly as much fun any more.
Of course since I'm a Bleach character, most of my friends are other Bleach characters and so my experience is limited to them. It's possible that other people from other animes and fandoms haven't forgotten how to rp correctly, but I doubt it.
Now, of course there is a time and place to act like yourself while on your rp account, but to do that you need to put "OOC" before you do. OOC means Out of Character. That's the time you are stepping out of character for a little, and commenting as yourself. Another place to be yourself is when you're having a one on one conversation with a friend and no one else can see the conversation, like on a private chat, or in an inbox. But other than that, since you are saying you are a character, shouldn't you act like them?
For those who are interested, here's a link to my rp account on Facebook: Arisawa Tatsuki