Step 4: What does all that DO - How does a fluorescent light work, anyway?
A fluorescent light is a gas discharge tube. It works a little like a strobe tube, and a little like an LED. Once it's running, it will happily allow very large electrical currents to flow through some ionized gas. To prevent it conducting so much power that it burns out or blows fuses, you have to limit the current with some sort of external circuit (this is the part that is similar to LEDs.) This is the main purpose of the fluorescent ballast. (The other function of the ballast is to get to that "once it's running" state. This can involve filaments, high(er) voltage pulses, and stuff like that.)
The picture shows a simplified fluorescent tube and ballast. You'll notice that the ballast is an inductor. This is because an inductor can act as a current limiter 'for AC current without actually using up any power the way a resistor (as used for LEDs) would. A neat trick. The current through the inductor (and thus the lamp, since it's a series circuit) is proportional to the AC frequency, and the inductance of the inductor. If you've ever seen the magnetic-only ballast from a standard fluorescent light, you'll have an idea how large an inductor is required at the 60Hz AC that comes out of the wall.