Although I don't want to spend too much time talking about exactly how a Tesla Coil works, I think that a rudimentary description is worthwhile given the amount of misinformation that exists on Instructables and elsewhere on the web.
Essentially, a Tesla Coil is a type of alternating current transformer that operates almost like any other (transformers are found in many electrical and electronic devices and are used to step up or step down the voltage of an alternating current signal). However, it relies on the principle of electrical resonance
in order to massively increase the voltage of the alternating current signal.
One comment misconception is that the primary circuit (capacitor and inductor) "amplify" the signal from the high voltage transformer and that the ratio of turns between the primary and secondary coil is then used to create a high voltage. However, this is not quite the case.
During each alternating current half-cycle, the transformer charges the primary capacitor until the voltage across it exceeds the breakdown voltage of the spark gap. At this point, the capacitor and primary coil are connected and momentarily form a series LC circuit. Because the capacitor has an initial charge from the transformer, the LC circuit will oscillate much like a stretched spring will move back and forth once it is released. In fact, the differential equation describing a stretched spring moving back and forth with friction is virtually identical to the one that describes an LC circuit with an initial charge on the capacitor oscillating with stray resistance in the wires of the circuit.
These oscillations can exhibit three different types of forms: overdamped, critically damped, and underdamped (second image). In the overdamped condition (high damping factor, ζ), the current decays without crossing zero, while in the underdamped condition (low damping factor), it crosses zero many times and oscillates before decaying. This last condition is the one we hope to achieve in our coil.
Once the circuit is oscillating, the rising and falling magnetic field around the primary coil will induce current into the secondary coil. The goal is to maximize energy transfer between the primary and secondary coil and minimize energy lost to heating as a result of stray resistance.
The secondary circuit also acts as an RLC network. Its impedance
, or resistance to an alternating current, will change as a function of the frequency that the primary circuit oscillates at. The third picture shows this relationship. If the frequency of the primary circuit matches that of the secondary circuit, then the amplitude of the secondary voltage will increase dramatically because the secondary impedance will be very low. Once the oscillations in the primary circuit have decayed, the transformer will switch polarity and recharge the capacitor, causing the cycle to repeat. This is similar to what happens when you try to force a sping to move back and forth; if you're not at the correct frequency, then it resists your push, but if you do manage to hit the right frequency, then even a small application of force can quickly increase the amplitude of its oscillations.
If you want a more mathematically rigorous explanation, be sure to take a look at this paper: http://tayloredge.com/reference/Machines/TeslaCoil.pdf
There's actually quite a bit more going on than I made it sound like, so consider taking a look even if you're going to skip the mathematics.
SAFETY WARNINGS (READ THESE):
That said, I want to give a few warnings to anyone who is considering this project. Tesla coils and other high voltage devices are extremely dangerous in the wrong hands and can easily injure or kill anyone who does not practice proper high voltage safety. I am not responsible for any accidents that may occur as a result of these instructions.
I also do not guarantee that your coil will work or that you will be satisfied with the results. Only attempt this project if you are willing to face failure on your first attempt and don't cut corners - if that capacitor has to be rated to a certain voltage or that wire has to be enameled, don't try to get an inferior product for less. It's better to wait and save up for the higher quality part than to end up with a pile of cheap, burnt out components.
Be sure to read the entire instructable and completely understand what you will have to do before attempting this project!