Steampunk is a sub-genre (branch) of Science-Fiction distinguished from all other science-fiction by its prevalent Victorian-era (1837–1901 AD) aesthetics whether in art, music, fashion, technology, or literature. Its defining characteristic is the steam engine, which gave way to the steam locomotive and its revolutionary railway system. A secondary characteristic is air-travel by balloon, or "airship".
Within Steampunk itself are two more, widely recognized sub-genres: "Post-Apocalyptic", which is usually set in the future and depicts mankind as having experienced a setback to 19th-century technology; and "A Future That Never Was", which is most often set in the past and is based on what Victorian contemporaries envisioned for the 20th-century. A third, and highly debated, sub-genre of Steampunk is Historical Purism; it's still science-fiction, but it ejects the wilder "punk" elements in favor of retaining a truly authentic Victorian flavor. In their own way, and regardless of whatever artistic medium, all three aim simply to revive the technologically romantic atmosphere first imagined by the novelists Jules Verne and H. G. Wells.
There are also variable grades of Steampunk. Some are impressive, some seem like Steampunk in name only. To be pollitically correct, you'll have to identify which is which for yourself.
Furthermore, some people who attempt to define "Steampunk" and struggle may fall to insisting that "Steampunk is everything, everything is Steampunk," due to this genre's immense versatility. However this does a serious injustice to the many other great subcultures of the world. For instance, "Cybergoth" kind of looks like Steampunk, but it's obviously different enough that Cybergoth is best described as Cybergoth. You don't have to lump them together under an umbrella term in order for them both to be cool.
As a final note: Steampunk, though inspired by European industrialism, is not to be confused with Euroboo-ism. A "Euroboo" is a person who wasn't born European, but rejects the culture and country of their birth because they hold a deep personal belief that all things European are superior. The genre of Steampunk can be localized to anywhere in the world at the artist's whim, no allegiance to Europe necessary because it's the science-fiction element that's important. For instance, "The Wild, Wild West".