Steampunk is a subgenre of Science Fiction (or sometimes Fantasy) in which, rather than digital technology coming into being, steam-powered mechanical machines became the normal form of technology.
Many of the devices in Steampunk are, of course, gear/steam/motor oriented, and some stories, like the Difference Engine, even have television-like screens that are powered by those elementary devices. Almost everything in Steampunk is an anachronism: a time that we could have had but did not, therefore society evolved differently, yet many devices still popped up, yet at a different time.
Steampunk technically stems from Cyberpunk (hence the name), being that it has the similar bent towards hopelessly technological worlds, though many Steampunk worlds are actually almost fairytale-seeming. A vast majority of Steampunk stories mirror Victorian England in culture, with men in top hats and women in those poofy dresses, although the more interesting Steampunk stories tend to stray away from Victorian settings.
Because of the bent towards Victorian England, many Steampunk stories (such as the Difference Engine) are also classified as Alternate History, a branch of Science Fiction and Historical Fiction.
In my opinion, Steampunk is very interesting to write in, because there is something extremely attractive about steam-covered cities full of factories and exotic machines, with a Star Wars kind of romantic feel (the settings, not necessarily the stories).
One thing that interests me in many of the Steampunk stories is how such things as consumerism were still invented, and mass production (popularized in c. 1900's) becomes overblown, like in the Diamond Age.
Some examples of Steampunk-like technology in our world:
Theremins
Tanks
Radios and Telegraphs
Tesla Coils and Van de Graff Generators
Zeppelins
Some popular examples of Steampunk:
Video Games
(to an extent) Final Fantasy IV/II
Final Fantasy V
Final Fantasy VI/III
(to an extent) Final Fantasy VII
(to an extent) Chrono Trigger
Rudra no Hihou
Steampunk Saga
Arcanum
Books
The Diamond Age
The Difference Engine
The Steampunk Trilogy
Celestial Matters
Many H.G. Wells and Jules Verne works