Tabletop games are games which don't involve physical competition (separating them from sports) or electronic devices (separating them from Video Games). It's generally agreed that a tabletop game must involve some physical objects, otherwise it would be a Parlor Game. A table is not required, however; most tabletop games can also be played sitting on the floor or standing up, but a table is usually the most convenient option.
There are many different categories of tabletop games, including:
And then there are lots of games that don't fit into any of these categories very well, from Mahjong to Connect Four.
Of course, the variety in gameplay is also enormous. Two aspects in which tabletop games can differ quite significantly are hidden information and random elements. For example, in chess, there is no hidden information and no randomness; everything is determined by the players' own actions, and everything in the game is revealed to both players. In Stratego and Diplomacy, there are no random elements either, but the amount of hidden information is huge, and deceiving your opponent(s) is an important part of gameplay. In the Game of the Goose, there is no hidden information, but the randomness is turned Up to Eleven – everythingis random, and players have zero control over the outcome of the game. Poker is a good example of a game that features both random elements (which cards you are dealt) and hidden information (your opponents' hands). Most games likewise have a little of both.
Recently, Wil Wheaton has promoted the board game hobby with his web series, Tabletop, where he and several 'geek celebrities' (famous geeks and people involved in cult nerd shows and projects, oftentimes both) play a variety of board games, from Settlers Of Catan to Ticket To Ride to the Dragon Age tabletop RPG campaign. It can be watched here as well as from YouTube, and has gone through two seasons at the time of this writing.
See Tabletop Game Tropes for tropes that apply to tabletop games and Game Tropes for a more general index.