What does a capacitor do in a tesla coil?
The tesla coil is a resonant transformer. The key to making them work is to generate a lot of power at a particular frequency to drive the primary side of the transformer. The basic way of doing this is to take a bunch of energy and dump it into an LC circuit. The LC circuit is made up of an inductor (L) and capacitor (C) in series and has a natural resonant frequency determined by the value of L and C. When you dump a bunch of energy into an LC circuit it wants to oscillate at it's resonant frequency. This generates the frequencies needed to drive the coil. So, to answer your question, the capacitor makes up half the LC circuit which generates the frequencies that drive the coil.