In physics, flux is defined as the flow rate of something (water, electromagnetic fields, etc.) through a given surface area. As Donald.McLean said, a capacitor is normally used to store electrical energy.
If we allow ourselves to enter the realm of movie physics, the term "flux capacitor" could be interpreted as something that accumulates and stores the flow of something for further use. Although this seems weird, knowing that we are talking about time travel, it could be that the flux capacitor stores the flow of time relative to the DeLorean and uses it to travel through time.
Donald.McLean has another great theory in that the accumulation of such flux (or flow of time) can be used to punch through some sort of time travel barrier. I believe this is at least partially correct (if not totally correct). In fact, the first movie itself seems to point this way. Although we don't see in the movie how the flux capacitor works exactly, Doc explains to Marty that the stainless steel body of the DeLorean has a direct influence on the "flux dispersal". Sadly, he is interrupted before he can finish his explanation.
Given how cages made of conductive material works (the steel frame of a car acts as a Faraday Cage, essentially), one could assume that the flux capacitor accumulates a certain amount "time flow" which it then releases through the body of the DeLorean as a means to counteract the effects of time travelling. And since the travellers are inside a protected area, they are immune to the adverse effects of such travel.
Without any word from the creators of the franchise, however, this explanation will remain pure conjecture on my part, although I feel it is faithful to the movies' general feel.
As for the point on lightning, Donald.McLean is also right in saying it is only used as a power source. The amount of energy needed to operate time travel in the BTTF universe is astronomical. In fact, 1.21gW (Watts being a unit of power, as in joules per second) is about the same power output of a normal nuclear reactor inside a nuclear power plant. In the future, Mr.Fusion household reactors have replaced the need for centralized powerplants and Doc mounts one on the car to replace the old plutonium reactor.