Capacitors are like super fast batteries. They can absorb power and release that power very fast. With the caveat they can’t contain much of it.
What are they used for does vary a lot.
If you have a signal going up and down: You use a capacitor that is just fast enough to charge with the peak of the wave and release when the bottom of the wave passes. The result is to flatten the wave. If the wave is just a power supply, you can use it to smoothen that power signal, killing dangerous peaks and filling voids. This is pretty basic. You can use it to smooth the power output of a transformer.
You can use it on a radio wave to “kill” noise on the signal. You can use a variable capacitor to tune the radio to a frequency by removing everything else. Basically, by adjusting the diode you adjust the frequency of the system.
Capacitors are pretty simple. The problem is that they can be used for a lot of tricks, and the equations to do that become quite complex quite quick.
Then there are more complex uses.
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