Tesla coils are a class of isolated resonant transformers invented by Nikola Tesla in the late 1800's. Tesla's vision for his coil was to resonate the earth and provide free power to everyone. This of course did not pan out; but the world got an awesome, easy to construct lightning machine out of his genius anyway.
The device consists of a primary pulsed LC drive circuit magnetically coupled into a secondary free wheeling LC circuit, the frequencies of both LC circuits are matched to provide maximum power transfer between the two circuits. The secondary can either be configured as a half wave or quarter wave resonator.
equation determining the frequency of an LC oscillator
A large part of the magic to the Tesla coil is a high amplitude damped sinusoidal current pulse, this is accomplished by discharging a capacitor into a low impedance. To perform this discharge some form of switch is required; typically a spark gap. Modern Incarnations utilize solid state switches such as MOSFETS or IGBT's.