Let's start with this introductory video that will tell you more or less how it works tesla coil followed by some text, and finally how to create your own coil

A Tesla coil is an electrical resonant transformer circuit invented by Nikola Tesla around 1891. It is used to produce high-voltage, low-current, high frequency alternating-current electricity. Tesla experimented with a number of different configurations consisting of two, or sometimes three, coupled resonant electric circuits.

Tesla used these coils to conduct innovative experiments in electrical lighting, phosphorescence, X-ray generation, high frequency alternating current phenomena, electrotherapy, and the transmission of electrical energy without wires. Tesla coil circuits were used commercially in sparkgap radio transmitters for wireless telegraphy until the 1920s, and in medical equipment such as electrotherapy and violet ray devices. Today their main use is for entertainment and educational displays, although small coils are still used today as leak detectors for high vacuum systems.
High voltage production[edit]
A large Tesla coil of more modern design often operates at very high peak power levels, up to many megawatts (millions of watts[46]). It is therefore adjusted and operated carefully, not only for efficiency and economy, but also for safety. If, due to improper tuning, the maximum voltage point occurs below the terminal, along the secondary coil, a discharge (spark) may break out and damage or destroy the coil wire, supports, or nearby objects.

First Coils
American Journal Electrician gives a description of one of the first Tesla coils, where an accumulator 15 cm glass by 20 cm glass is rolled between 60 and 80 turns of wire of copper highest possible percentage of No. 18 AWG. Within this consistent primary coil it is at about 8 or 10 turns of number 6 AWG copper wire and the assembly is immersed in a vessel containing linseed oil or mineral oil.





Tesla coil schematics     

Typical circuit configuration
Here, the spark gap shorts the high frequency across the first transformer that is supplied by alternating current. An inductance, not shown, protects the transformer. This design is favoured when a relatively fragile neon sign transformer is used.

Alternative circuit configuration
With the capacitor in parallel to the first transformer and the spark gap in series to the Tesla-coil primary, the AC supply transformer must be capable of withstanding high voltages at high frequencies.
Tesla experimented with these, and many other, circuit configurations (see right). The Tesla coil primary winding, spark gap and tank capacitor are connected in series. In each circuit, the AC supply transformer charges the tank capacitor until its voltage is sufficient to break down the spark gap. The gap suddenly fires, allowing the charged tank capacitor to discharge into the primary winding. Once the gap fires, the electrical behavior of either circuit is identical. Experiments have shown that neither circuit offers any marked performance advantage over the other.

However, in the typical circuit, the spark gap's short circuiting action prevents high-frequency oscillations from 'backing up' into the supply transformer. In the alternate circuit, high amplitude high frequency oscillations that appear across the capacitor also are applied to the supply transformer's winding. This can induce corona discharges between turns that weaken and eventually destroy the transformer's insulation. Experienced Tesla coil builders almost exclusively use the top circuit, often augmenting it with low pass filters (resistor and capacitor (RC) networks) between the supply transformer and spark gap to help protect the supply transformer. This is especially important when using transformers with fragile high-voltage windings, such as neon sign transformers (NSTs). Regardless of which configuration is used, the HV transformer must be of a type that self-limits its secondary current by means of internal leakage inductance. A normal (low leakage inductance) high-voltage transformer must use an external limiter (sometimes called a ballast) to limit current. NSTs are designed to have high leakage inductance to limit their short circuit current to a safe level.

AND WHY NOT MAKE YOUR OWN TESLA COIL?

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