Roleplaying is basically the act of playing the role of your character beyond running about hitting things. Developing their personailty, backstories etc and acting them out based on those.
(Best way I could put it. :\ )
Sam: "My mind is a swirling miasma of scintillating thoughts and turgid ideas."
Max: "Me too."
Stuff
pointless just about covers it.
acting as the character, and the character being a 'real' (in game universe terms) entity, not a player behind them.
best way to put it is the occasional moronic comment you see made to RPers: "lol y u all talk leik npcs???!"
Where characters act in a sort of storyline, for example 2 friends who have accounts in the game pretend their characters are brothers, and this may be a converstation:
Guy 1: "Hey bro"
Guy 2: "Hey"
Guy 1: "You saw that guy Morado yesterday? He's a jerk"
Guy 2: "Totally agree"
Guy 1: "I gotta run off to the grocers, see ya"
Guy 2: "Alright, bye bro"
And that conversation is just basic Rokeplaying, some players will have storylines like [player] kills [player] and [player]'s brother avenges [player] and manages to kill [player] (although that's a bit far)
As others have said: they talk like real people not just general normal chat you get when playing the game. It's a bit like real life. (I'd better stop, this is so reminding me of Second Life)
Basically, roleplaying (RP) is acting/pretending.
It could be used in psychotherapy; for instance in relationship counselling. A married couple could 'roleplay' and pretend they are angry with each other (or whatever situation they want to deal with) and then act out how they would react to one another to better deal with when it really happens... yada, yada...
But the thing you ask about, I guess, is the RPG (roleplaying game).
Primarily, it's pretending, acting and talking as if you where your character him-/herself and not just a bloke sitting infront of a computer playing a game.
The most classic version is the table-top RPG where you usually sit around a table with some friends and have sheets of paper that describe your characters' abilities and mostly talk about what happens. You often use dice to determine the degree of success to get some randomness to it.
One person is the GM (game master) that tells the story that the others play out and he also controls the actions of the NPCs (non-playing characters).
The more modern and increasingly popular version is the online game, like City of Heroes. It's more or less the same as the table-top RPG, except that the GM, NPCs, story etc. is already done with the programming of the game.
- A non-RPer doesn't give a damn about acting. He is a person sitting by a computer, playing a game. He doesn't pretend anything else.
- A hardcore RPer acts his character as soon as he comes online. Everything he does and says is according to how his character would react.
- A casual RPer acts out his character when he feels like it, has inspiration for it or perhaps only when his other RP friends are online. (It can be a waste of time roleplaying if noone else cares about it.)
A non-RPer in an online game could say:
"OMG!!1! lol mum wants me 2 clean up in teh kitchen brb lol" And your character stands completely still in the middle of the streat with big <AFK> letters over his head.
A roleplayer could in the same situation instead come up with an explanation why his character disappears or is standing still, being silent for a few minutes and respond:
"Excuse me, but there is something else that requires my attention. I will come back to you in a minute!" Then do a /bow emote and run off to a 'safe' place and log out/whatever.
IF you dont' like RP, it's up to you. But roleplaying can be a fun way to give flavour to a game that perhaps doesn't have much more to offer. If you sit around with fifteen lvl 50 characters of each type, everyone fully slotted with Invention Sets or Hami-Os and still have 100 million influence left, you aren't perhaps so interested in leveling yet another character.
Instead you could start having some fun with the ones you have without doing the monotonous "Defeat 15 skulls. Defeat 20 Skyraiders. Defeat 10 Circle of Thorns" that can be quite mind-numbing.
Computer games are mostly just an advanced way of wasting/spending time, so why not do it with another approach that can be quite entertaining from time to time?
Ok, sorry for the massive wall of text, but I hope that this gave you some answers atleast. (If you didn't fall asleep before reading though, that is)
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This is Roleplaying.
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NO, that's a retard trying not to end in the bunk of whom ever has the most Cigarettes. it has nothing to do with role playing.
it looks like he was roleplaying the elf to me...
@Rooks
"You should come inside the box... Then you'll know what I mean."
Thats right he was, which is why it was used. Even if that is someone who could be said had been taken over by his character, but as far as i see he was just trying to used that to get himself out of trouble, which is the worse idea he could of thought of. 
That's what happens when you roleplay a transvestite elf. Transvestite elves have a deep-set craving to rob lingerie for their own twisted ends.
We, however, roleplay superheroes, so we're gonna run around the town center in spandex punching muggers in the face. 's all good.
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what is Roleplaying?
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Roleplaying is like Marks and Spencer Gaming.
*'Bow, bow-wow' M&S music*
This isn't just a Super Group.
This is a hand crafted Golden Age Super-Hero Justice Force, with a collective backstory of over 10,000 words per character, bristling with ripe and juicy story, intrigue and monthly character driven plot-lines.
This isn't just a Fire Blaster called "teh human scorch1"
This is the "Fantastic FireStorm", complete with a family tree, origin saga, Arch-Nemesis, social acquaintances and battle-scars aquired from the First and Second Rikti Wars.
This isn't just Playing. This is Role-Playing.
It takes Chaos to move the world to Action.
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This is Roleplaying.
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NO, that's a retard trying not to end in the bunk of whom ever has the most Cigarettes. it has nothing to do with role playing.
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No, I'm afraid to say that's definitely RolePlay. =D
KlerRP, Kinky Lingerie Elf Retard Role Play.
It takes Chaos to move the world to Action.
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This is Roleplaying.
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and these guys roxxors teh PVP!
that guy is 1 part RPer, 10 parts delusional nutjob.
most real RPers are the other way around
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This is Roleplaying.
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Damn WoW RPers 
I cant possibly put it better than Wiki, as that covers the subject perfectly.
<ul type="square">[*]Act out a character.[*]Come up with an intresting character background.[*]Give it a personality, depth, feelings. That could be diffrent to your own.[*]Develop your character through roleplaying, be it Forum Roleplay, ingame Roleplay, IM, etc. Theres so many diffrent ways.[*]Do what your character would do, not what you would do. Although in some cases, your character might have a similar personality to yourself![/list]
Thats a very quick summary, some people who've already posted have hit the nail on the head. 
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This is Roleplaying.
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and these guys roxxors teh PVP!
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Awwww what's up? Truth hurts a lot?
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This is Roleplaying.
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and these guys roxxors teh PVP!
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Awwww what's up? Truth hurts a lot?
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and that just goes to show me that your original post was just an attempt at trolling
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This is Roleplaying.
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Damn WoW RPers
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Well... If you had read the article you'd know that it had nothing to do with WoW, but instead the old classic RPG Shadowrun.
WoW RP:ers can be annoying too, but this time they are innocent.
Stupid question, but i really don't know so what is Roleplaying?