Technical data:

tesla coil running 
In a simple model, it can be described as two inductively coupled resonance circuits. The primary cap is charged by the primary transformer. When the spark gap fires, the primary resonant circuit is closed and starts to oscillate. If the circuits are tuned to the same resonance frequency, energy is transferred from the primary to the secondary circuit (and back and so on periodically) in the form of increasing oscillations. The maximum voltage in the two circuits depends on the relative values of capacity and inductance. In the case of a tesla coil, the secondary inductance is high (many turns) compared to the primary inductance (few turns), causing the secondary voltage to be higher than the primary voltage by a factor of maybe 20.