The circuit is called a Slayer Exciter. It is a hand-wound air-core transformer (it took me about two hours to wind the transformer; the red cylinder in the picture is actually about 400 turns of 34-gauge magnet wire) capable of generating a high voltage and a very strong electric field capable of illuminating a fluorescent light bulb. There are lots of online how-tos floating around which go into great detail about this "poor man's Tesla coil" so I won't go into great detail here. Basically, it's a very simple oscillating circuit consisting of a transistor, two diodes, and some resistors that can turn a low voltage DC source into very high voltage AC. Here are some pictures!
Some prototyping, just to make sure it all works:
Putting it all together on a piece of wood, as I am wont to do with electrical things:
And, on an unrelated note, perhaps the most random two things to be purchased at one time from Amazon. A 14-ounce tin of almond flour for gluten-free baking and a Lieutenant Commander Geordi Laforge action figure to make sure I have the nerdiest cubicle in the office:
MY TRICORDER DETECTS NO WHEAT, CAPTAIN. IT'S A MYSTERY!