I am not responsible for any bodily damage or property damage that is a result of one building or using this circuit. High voltage can be dangerous or lethal when used incorrectly, and it is up to the user of the circuit to use high voltage safety rules and maintain their own safety at all times.
Flyback transformers are some of the most commonly used high voltage supplies by hobbyists- mainly due to their relative simplicity and how easy they are to find. Every single CRT (big, heavy kind) TV that people are throwing out now has one, the bigger the TV the bigger the transformer.
But unlike many transformers that you find in electronics, which are designed to handle standard 60 hertz AC power, and are step down transformers, flyback transformers operate at much higher frequencies- around 16 khz or sometimes higher. Many modern flybacks also put out DC voltage. Older flybacks had AC power output, making them the ones you want. The AC flybacks can operate at higher power, since there are no rectifier/multiplier circuits built into them like the DC flybacks.
That being said, DC flybacks are easier to get. So for this project they are recommended. Make sure that the flyback transformer you are using has an air gap. This means that the core is not a loop all the way through, but is rather shaped like a C with the gap being about a mm thick. Almost all modern flybacks have an air gap, so if a new on is being used, checking is not necessary.
Now, onto the circuit. I am going to say right now this is not my circuit. I found it on the 4hv.org forums here. This Instructable is intended to take much of the theory presented in the thread and condense it into a quick how-to for beginners.
This circuit uses the venerable 2n3055 transistor, which is loved/hated by flyback driver builders. Loved because it is so easy to get (even Radioshack sells them) and hated because they generally stink. They are prone to burning out, sometimes spectacularly. That being said, this circuit works incredibly well with it.
The bad reputation for the 2n3055 comes from it being used in the usual single-transistor flyback driver, which puts a lot of stress on the transistor. In this circuit, a few components have been added which dramatically increase the power that the circuit can put out. The theory of the circuit is discussed later in the Instructable.