As a simple explanation, a Tesla coil works by utilizing the principle of resonant rise. There are two coils in a Tesla coil, the primary and the secondary (the much longer, cylindrical one with many turns). Both of these coils are tuned to the same frequency. A high voltage power supply (neon sign transformer) charges a capacitor that discharges its energy into the primary coil through a spark gap. When the energy discharges into the primary coil, it creates an oscillation that excites the secondary coil. As a result, the voltage increases strongly and discharges through the toroidal terminal. The blue/purple "lightning" that the observer sees is the ionization of the air, which consists of 70% Nitrogen, and thus is responsible for the color. With different gases, other ionization colors may be formed.
Running off of a disassembled neon sign transformer, the power supply for this coil is 12,000 volts @ 200ma. The secondary coil is 4in diameter with a 22in winding length. Its resonant frequency is about 145khz. The top load is made with aluminum ducting. The capacitor is a Maxwell 0.035uf @ 35,000 volts.
The spark gap is a simple RQ gap that dissipates the heat relatively well.
My larger Tesla coil will be powered off of a 14,700 volt 700ma, Power Distribution Transformer, or "pole pig".
Thank you for watching.