Obviously it is impractical to use capacitors to store any significant amount of power unless you do it at a high voltage. The difference between a capacitor and a battery is that a capacitor can dump its entire charge in a tiny fraction of a second, where a battery would take minutes to completely discharge itself. They contain big capacitors that can, potentially, kill you with the charge they contain. That's what a flash does. Big lasers use this technique as well to get very bright, instantaneous flashes.
Capacitors can also eliminate ripples.
If a line carrying DC voltage has ripples or spikes in it, a big capacitor can even out the voltage by absorbing the peaks and filling in the valleys. A capacitor can block DC voltage. However, any alternating current signal flows through a capacitor unimpeded. That's because the capacitor will charge and discharge as the alternating current fluctuates, making it appear that the alternating current is flowing. One big use of capacitors is to team them up with inductors to create oscillators .