[This deserves a proper treatment. I will just summarize based on what I know.]

Probably the most standard form of the RolePlayingGame. The first games that were called "role playing games" were the pen and paper variety, unless maybe you count the version played in the bedroom. The latter might be beyond the scope of this site. The first p&p RPG to gain popularity was Dungeons_&_Dragons?, and it is possibly still the biggest (if you lump it together with it's sibling, Advanced_Dungeons_&_Dragons?). It was invented by Gary_Gygax? and David_Arneson?. Gygax and Arneson developed the first pen and paper RPG as an extension to their miniatures game, Chainmail?. (Read more about the beginnings of D&D.) Countless others have been and continue to be produced.

The principle is that a game master (GM; or Dungeon Master in D&D) oversees the game for a group of players. The GM knows what is going on behind the scenes; he (or she) is responsible for managing the milieu and presenting the world to the players as their characters see it. Common equipment includes player and GM manuals, maps and guides prepared in advance by the GM, dice to allow for randomness in events, and possibly a scale model and/or miniatures to aid in simulated combat. The game world and the situation the player characters (PCs) find themselves, the milieu, is wide open. As these are quite a lot of work to design, they are often sold as accessories to the rule books.

Like other forms of RPG, a typical game involves the GM introducing the characters into the current situation in much the same way as an author introduces his characters -- and readers -- into an adventure story. There is some plot element which affects them, usually a crisis or an opportunity. The PCs must decide what to do. Their actions can be divided into colloquial and governed actions. Colloquial actions are where most of the role playing comes in, with things like wandering around and talking to non-combatant NPCs. The governed actions are those wherein the PCs attempt to exert themselves upon the world and its other inhabitants: through combat, thievery, spell casting, persuasion, and other skills. PC (and NPC) skills are rated numerically, usually with charts or percentages. Actions are governed according to the rules set out by the game system, which often accounts for the chance of success, the order in which they occur, the time they take, areas of affect, and limits on their rate of repetition. Time is very important in many systems, and is measured solely based on what the characters do, not on real time. The GM must account for all of this -- it is a very demanding role.

The bulk of a typical scenario involves exploration and combat. Part of the motivation of playing is to "beat the game", as it were, and this is rewarded by improving the lot of the PC. The standard measure of success is in experience points, or XP, which converts into better skills. The PCs gain spoils, as well, as any good mercenary expects -- most characters in the scenarios of RPGs are mercenaries of some sort, but usually they call themselves adventurers. The adventurers enter some dangerous place under the control of the players, encounter monsters of various types, and attempt to survive and overcome the dangers and make off with the loot. In the midst of this, they are creating a story, and the world the characters live in should change as a result.