Q: What is a Tesla Coil?!

The non-technical answer to this question is:-

A: A Tesla Coil is a device which generates artificial lightning.

The technical answer, as far as my limited understanding goes, is:-

A: A Tesla Coil is a resonant air-cored transformer. The primary winding forms part of a tank circuit (tuned L-C oscillator) which is loosely coupled to the secondary coil. The secondary coil has one end connected to earth, and the other end has a capacitive load (a metal sphere or other smooth object), making another L-C resonant circuit. The resonant frequency of the secondary is the same as the primary coil circuit. Because the two coils are only loosely coupled (no iron core, just air) the voltage induced in the secondary winding is allowed to ring up to a very high voltage, more than can be accounted for by the turns ratio of the transformer. The strong electric field generated around the capacitive load terminal (sphere) causes lightning-like streamers to break out into the air.

The output voltage can be hundreds of thousands of volts, although a true measurement of this is difficult! The tank circuit can operate from a supply of 8 to 15 kV (50/60Hz AC), derived from normal mains voltages (110/220/240 VAC) with the aid of a high voltage transformer.