There's been a lot of discussion on how the flux capacitor functions, but let us not forget the circumstances around its invention. On November 5,
1955 Doc Brown was hanging a
clock on the bathroom wall while standing on the
toilet. The toilet-top was wet, however, and he slipped and hit his head on the sink. When he regained consciousness he had a vision of a device that would make
time travel possible. He drew a quick sketch of a Y-shaped assembly of tubes that he dubbed the "flux capacitor".
Doc Brown spent the next thirty years of his life and his entire family fortune to realize the vision of that day. The Doc decided to build the flux capacitor and time circuits into a DeLorean ("The way I see it, if you're going to build a time machine into a car, why not do it with some style?" he said in 1985). The flux capacitor in the DeLorean is located behind and to the side of the driver's side seat. There are small warning labels glued to the device's case that warn not to stare directly at the flux capacitor while it is in use. This iteration of the flux capacitor requires plutonium to operate (or, in a pinch, the 1.21 jigawatts* that a bolt of lightning can provide (SEE: Back to the Future)) The car was refitted with a Mr. Fusion device in 2015 that allowed the flux capacitor and time circuits to operate from a fusion process that was fueled by garbage (SEE: Back to the Future II).
The DeLorean and its flux capacitor were destroyed when the car was hit by a train in October 1985. Doc Brown was trapped in 1885 at the time (SEE: Back to the Future III), however, and over the following years he constructed a new time machine out of a train. The flux capacitor on this train is located on the exterior front of the engine and glows red in contrast to the DeLorean's white capacitor. This version of the flux capacitor, being designed from 19th century parts, runs on steam.
The flux capacitor is one of science fiction's most memorable devices and it's been referenced in a number of other sci-fi franchises including Star Trek: The Next Generation (episode: "Rascals") and Mystery Science Theater 3000.
* gigawatts
References:
Back to the Future Trilogy DVDs