This very site you are currently browsing is first and foremost a site about role-playing games, a jumping-off point for GMs and Players seeking to liven up their scenarios and characters.
What is a role-playing game? It's a hobby that has elements in common with improvisational acting, collaborative storytelling, radio drama, playing "cops and robbers" as a kid, and board gaming. Most, but not all, role-playing games use some variation of the following format:
One participant, known as the Game Master, Dungeon Master, Referee, or Narrator, and referred to below as the GM, creates a setting or scenario. He or she guides the other participants (known as Players) through it, much as the narrator of a story does. The Players each has a single role or character they play, often referred to as their Player Character. Unlike the actors in a play, players in an RPG are not bound by a script - their choices and decisions are an important part of the medium. Amidst bits of voice-acting, they indicate to the Game Master the actions their characters will do. Instead of having a single role, the GM plays numerous bit parts (called NPCs) and describes the setting. He also moderates the consequences of Player Character's actions, and describes to the players what their characters see or experience. Typically, a set of rules define what sort of things a character can do, and their odds of success at those tasks. This may involve the rolling of polyhedral dice, or some other method of randomization. Random methods mean that success or failure is never guaranteed, and help maintain dramatic tension about the uncertainty of how the plot will unfold.
The first role-playing game was Dungeons & Dragons, published in 1974. A great many other games have been designed since then, exploring a multitude of genres, and utilizing a variety of different mechanics. Endless variations of the above format exist. See Indy RPG, LARP, and Role-playing Game Theory for other takes on how to roleplay.