Tabletop RPGs (also called Pen-and-Paper RPGs) are roleplaying games in which a person takes on the role of a fictional character, as supervised by a game leader, often called the GM or DM (game master and dungeon master, respectively). Most of the time, it is for the purpose of playing out fantasy adventures, such as slaying demons or rescuing princesses.
Tabletop RPGs focus on combat, and often contain volume after volume of books which explain and expand upon rules, and add optional features as well.
Tabletop RPGs tend to have a medieval-setting, but include classical fantasy monsters such as dragons, and paranormal events such as spells and magic. Most tabletop RPGs are based in Tolkein lore, including the presense of elves, dwarves and hobbits.
Typically, when playing a tabletop RPG, you'll design your character in terms of their race, class and special abilities.
Racial choice (human, elf, gnome, etc.) usually sets the tone for your characters behavior. Dwarves are obnoxious and beer-crazy, elves are haughty and frivilous, and so on. Racial choice also often affects your statistics, such as how much damage you deal, how fast you can move, and so on.
Character class is basically their profession, though it is, again, combat-oriented. You won't see a Farmer class, but you'll often see archetypes such as Warrior, Mage and Theif.
Special abilities are those outside of race and class choice. They are optional abilities which either modify the way your character behaves in combat, or adds some stastical augmentation. In D&D, these are Feats and Skills. In Vampire: The Masquerade, this is tantamount to your Clan.
While the popular cliché would depict players of tabletop RPGs as losers and social outcasts, due to the most recent crop of fantasy-based major motion pictures, more and more demographics enjoy this passtime. This is made easier by the internet, as people can play in relative anonymity and across the globe.
Dungeons and Dragons is, perhaps, the most famous of all tabletop RPGs.