That's the most difficult question ever, though at the same time the answer does sound ridiculously simple.
Role-playing means just that. You pick a role and play it. You pretend to be someone else. You make believe.
If that's not exact enough, picture this: Consider COH a story rather than a game. A play, or a movie, or a novel -- or most obviously a comic book. Every character in the game is a character in the story. The things they do in game are what would be written in the script of the story. Main difference to a written story is though that you're in control of your character's actions but ONLY your character's actions -- you never know how someone else might react to you. Maybe a more clear term for RP is 'cooperative storytelling' cause that's what it boils down to. Several people are getting together and create a story together, each of them representing one character in the story.
The core idea is that the character you play in the game is not simply an abstract representation of yourself but an autonomous (albeit fictional) person. He or she is a guy with a life in Paragon City. They get up in the morning, probably use the bathroom etc. They don't discuss 'buying DOs with Influence' because these are abstract terms we use which they wouldn't be aware of in their world. They do things that have no real point in game (ie. don't give XP or make your character better) simply because it'd be something they'd do based on their personality.
Boiling it down, RP in COH would mean to play your COH character as if they were a comic book superhero rather than just yourself, playing a game. Maybe you'd just care to stop for a moment and consider how you'd act if you yourself had superpowers. You wouldn't be running around blasting Hellions all day either, I figure. There's nice superheroines to flirt with, celebrities to have your photo taken with or maybe you want to start your own super-powere boy band? Of course, you can't actually do some of these things in the game, but you can pretend you did them or plan to do them. It's not for everyone, but give it a try and see if that kind of approach adds to the fun factor.
