A Tesla Coil is essentially a very large oscillator. When the primary side of the coil oscillates at the natural frequency of the secondary side, resonance is achieved. This is a fundamental concept that is used in all Tesla Coils and other resonant transformers (such as the ones found in many switch-mode power supplies, and CRT television sets). The Vacuum Tube Tesla Coil that I detail here uses a configuration known as an Armstrong Oscillator.
In the standard model of a transformer, there are two coils, a primary and a secondary coil. Currents are usually induced from the primary coil to the secondary coil (although the opposite sometimes happens, usually with destructive results), this is a concept that we will not go over now, if you are unfamiliar with it, then this is a good place to become acquainted: http://en.wikipedia.org/wiki/Transformer.
However, an Armstrong Oscillator works by introducing a third coil, called the feedback, or sometimes "tickler" coil.
Currents are not only induced from the primary coil into the secondary coil, but also into the feedback coil. This feedback is then used to turn off the oscillator by blocking current from flowing into the primary coil. However, when the primary coil is turned off, current is no longer induced into the feedback coil, and it no longer blocks current from flowing through the primary coil. In this way, the cycle repeats indefinitely, until it is interrupted, or the power is switched off.
The basic schematic for an Armstrong Oscillator using a vacuum tube is given in the first picture. (This picture is from Steve Ward's site: http://www.stevehv.4hv.org/VTTCfaq.htm, you can read more about VTTC operation there)