Role-playing: One or more participants adopts a character or persona and performs or designates that character or persona's actions. Sometimes one or more participants will adopt the role of the world at large, and dictate its actions in response to the character(s). There is also sometimes a universal method of task resolution which may further define, in whole or in part, the world, to the degree that the world is fictional.
Role-playing game: A system of role-playing which includes goals for both the characters and participants. Goals for the characters vary; the mutual goal for participants is entertainment.
I think that pretty much covers it (you can fit LARPing, psychotherapy, and even group storytelling into there, although it's most suited for table-top gaming.) Rip it apart if you find a situation where it doesn't work
One thing I find kind of curious is the importance of task resolution to most role-playing games... Granted, you can do without it, but it would be hard to take that kind of game "seriously" -- which is to say, somehow task resolution aids in the suspension of disbelief.
I can't say, really, why this is: You don't have a mechanic for task resolution in movies, books, et cetera (although it could be argued that genre conventions can behave as a system of task resolution, but genre isn't a necessity in those forms, either)... Any thoughts on this?