Tabletop Wargame - a genre of games which have evolved over the years to encompass more than just wargames played on a table top.
The wargame has evolved since its introduction as a civilian hobby in the early twentieth century. Originally introduced by H. G. Wells about the time World War One began, Little Wars was quite popular and used small tin soldiers, a common children's toy at the time. As time passed, offshoots developed and became popular in their own right. One was the boxed military boardgame, introduced in the 1950's. Whether found in a magazine (Strategy and Tactics) or produced by a company (Avalon Hill), it was a cheap way to get into the general wargaming hobby, as the army was a collection of cardboard counters instead of expensive, hand-painted miniatures. It was also easy on space, as you could conceivably play it on any level surface (usually the kitchen table).
Another started out as a set of fantasy rules for use with traditional wargames. It appeared as an appendix to the game Chainmail, by Dave Arneson and Gary Gygax. This 'interesting afterthought' later became a full-blown game type of its own, the classic role-playing game known as Dungeons and Dragons.
Tabletop wargames began to expand in the early 1960's as military recreations of battles from World War Two became popular. Metal military vehicles, made of pewter or lead and cast in an established 1/285 scale, were fielded on a 4' x 4' table filled with sand. This sand table could be modified to represent nearly any kind of terrain, just as the originals were for the Prussian Army. Small buildings were sometimes made out of cardboard, along with toothpick fences, trees and so forth, to add a note of realism to the proceedings.
Sand tables are no longer in wide use (the last one I personally saw was in the back room of the American Eagles hobby store, North Seattle, in 1979). They have been replaced by solid wood tables measuring either 4' by 4' or (more commonly) 4' by 8', which feature green carpet material for a bed. Terrain is custom-made for these tables out of wood, styrofoam and resin-cast models. These features are stored alongside (or even under) the table, for use when a game is set up. The crafting of realistic terrain is popular and easy, and in recent years has become a hobby unto itself, much like the miniatures which are used in and around it.
The games themselves have evolved as well, gradually focusing more and more on the use of fantastically detailed - and expensive - miniatures. Such miniatures are no longer limited to metal casting, either - mixed media is the most popular form to appear in recent years, combining inexpensive plastic forms with pewter detailing. The collecting of these miniature armies, and the unique paint schemes applied to them, form yet another distinct hobby.
Some games have miniatures which are fantasy or science-fiction based, and have parts which can mixed and matched according to what type of unit you are fielding (Warhammer and Warhammer: 40,000). Others use miniatures which are faithfully cast along the old 1/285 scale guidlines (Classic Battletech). Still others are sold prepainted right out of the box, and are collectible (Mechwarrior: Dark Age, Mage Knight and Hero Clix). And some even consist of pieces you punch out of a plastic card (Pirates of the Spanish Main).
All share a common theme - a large table exclusively designed and reserved for this type of gaming; a set of variable terrain which can be quite simple or very ornate; and the extensive used of miniatures to represent the armies which each player will field.