A jetpack is a fundamentally fictional invention, although as we'll see later some rather impractical models have been constructed. Although it is difficult to say who first thought up the jetpack, what with it being essentially an idea rather than an actual thing, it's reasonable enough to presume that it could not have been conceived too far before the widespread use of rocket power. Although known in Western warfare since its introduction from the East in the thirteenth century, the rocket was something more of a novelty in European warfare than anything else, not unlike the gun when it was first introduced. It really entered the modern age of rocketry (that is, it began to look and act more like a space shuttle than a bottle rocket) with the advent of the German V-2 rocket during World War II.
This also places the jetpack's first appearance right around the time in which science fiction was really gaining legitimacy and relevance within American culture, namely the 1950's. At the same time as America started using terms like "The Jet Age" and "The Space Age" to optimistically describe the time of technological advances and social change that they hoped would somehow fire us all into the next millenium at breakneck speed, the constant threat of nuclear annihilation led many young authors to ponder the what-if's that their new postwar world entailed. Both of these ingredients hit one another at the right time, and since then it's been rare to see either jetpacks or science fiction apart from the other. In fact, when one passed on into the realm of camp, the other one did as well, which is why they can both now be seen on the consciously campy Futurama every week.
The archetypical jetpack is typically an amalgam of a rocket and a backpack worn, obviously enough, on its user's back. Any jetpack worth its salt is a tool for flying, first and foremost, and the most common method of becoming airborne is via some sort of flames or other force, such as the Jetsons' little force-ring things, being projected down and to the rear (which to me always seemed like a great way to light one's ass on fire). This is not to say that this is the only way of imagining a jetpack. After all, its fictionality does nothing if not inspire reimagination. One of the more innovative designs I've seen belonged to these two very obscure DC Comics superheroes from the '60's who lived in a shrunken Kryptonian city-in-a-bottle in Superman's Fortress of Solitude. These two, whose names have slipped my mind in the ten years since I read the comic (/msg if you remember), wore AA battery-sized cylinders mounted on rods that stuck out over each hip. These little cylinders shot fire downward, allowing them to fly around the shrunken alien burg righting tiny little wrongs.
The most prominent place the jetpack has ever found in any work to my knowledge is in The Rocketeer, which is both a recent movie (1991) and a comic book. Although it's been some time since I've seen said movie, I'll try to recollect it here as well as possible. The young hero, one Cliff Secord, played by Bill Campbell, stumbles onto a jetpack. The movie then trots out all the old superhero movie conventions such as heroic self-doubt, poor superpower control, a love interest, and gangsters. At the end, though, the Rocketeer hero and the gangsters both team up to defeat some Nazis who I think wanted the jetpack back for themselves. The jetpack pretty much makes the movie here in the sense that it's really the focal point of all of the action. Without a jetpack, there's really no reason to make The Rocketeer in the first place.
Although the jetpack is fundamentally a fictional invention, there are those who think that the idea of personal flight devices is pretty neat. One such group, Powerhouse Productions, Inc., has constructed an actual working jetpack, which they call the Rocketbelt. They have a couple of pilots, and you can apparently hire these pilots to do appearances with the jetpack at your corporate retreat, celebration, or stratospheric bar mitzvah, according to the website. Here's an amusing excerpt from their website:
"Carried aloft by a thunderous blend of science and technology,
The Rocketman captures the imagination of audiences worldwide.
The Rocketman is a great way to endorse any product.
The
creation of your own character can really enhance and personalize
your Rocketman Performance.
The Rocketman can ensure your message will rise above the rest
and thrust you to the top!"
Although I'll probably be chuckling about "a thunderous blend of science and technology" for years to come, I have seen this jetpack twice on television, and I have to admit that it's pretty cool. This is especially true when they're flying it through a rollercoaster loop at the same time as the train is going around it. Although the item itself looks quite goofy indeed (http://www.rocketmaninc.com/images/Egypt-2.jpg, if you don't believe me), I'm not going to tell you I don't want one. Personal flight seems like an incredibly cool prospect, and I guarantee I'd buy one if I had the money.
They'd have to figure out a way to not char my ass off, though.
Sources: http://www.invent.org/book/book-text/46.html,
http://www.ksc.nasa.gov/history/rocket-history.txt,
http://www.rocketmaninc.com/index2.htm,
http://us.imdb.com/Title?0102803