A capacitor is a kind of battery, cept' it can almost instantaneously discharge and recharge, generally used for high output amps driving subwoofers.
A crossover basically seperates the sound signal coming from the source, ie cd player, tape deck, etc. it helps promote a cleaner sound by sending the appropriate frequency of sound to the correct speaker, example a two way crossover usually comes with component systems. it splits the frequency range in half usually around about 1000-1500 hz unless you go with expesive brands/models. it will send everything below this cutoff to the woofer or larger speaker and the upper range to the tweeter. hope this helps