Roleplaying is essentially the children's game of "let's pretend," played by adults.
It's like Cops & Robbers, only there are rules to avoid the "I shot you!" "No you didn't!" arguments. It's sort of a cross between improvisational theatre and interactive story-telling. In Japan they're called "table talk games," which pretty much describes how they're played.
Roleplaying was first popularized by the game Dungeons & Dragons, but there are any number of other games available now (Call of Cthulhu, HeroQuest, Pendragon, Prince Valiant, RuneQuest, Star Wars, just to name a few I like).
Types of Games
I usually play with a bunch of people (4-6 works best) sitting around a table. One person is the gamemaster, who is responsible for describing the story's setting, and more or less guiding the direction the story goes.
Everyone else has a sheet of paper with notes on the imaginary character whose persona they're assuming, and dice to help determine success at difficult tasks (i.e. to see if you can climb up the scree, roll dice and compare it to your character's Climbing skill).
The same sort of thing can be done by mail (paper or electronic). The number of players can be as few as one, or a single game can have a dozen or more.
And there are live action games, where you walk around (often in costume) and interact with other characters. These tend to have fewer rules and less combat, and work best with dozens of players.
The Structure of Games
Most of the games I'm involved with are ongoing. Each gaming session is one story. The next session picks up where the last one left off, with the same set of characters becoming older, more experienced, and more powerful as the game continues. Due to the hobby's wargaming roots, each session is usually called a scenario (some call it an episode), and the continuing series is a campaign.
Another definition of roleplaying, and another, and yet another.