A flourescent light works because as the cathode is heated, it thermionically emits electrons which undergo collisions with vaporized Mercury atoms. As the Mercury atoms de-excite, they release a UV photon. This photon then interacts with the phosphor on the wall of the chamber causing it to emit visible light.
They require a ballast because they have negative differential resistance. This means that as more current is applied to the fluorescent tube, the resistivity actually decreases. This is dangerous because there is no limit to the amount of current running through the tube.
A ballast is connected in parallel with the fluorescent tube to limit the current running through it. There are many kinds, but the simplest is an inductor placed in series with the fluorescent lamp and the starter.
The starter is a device that provides a voltage spike to the tube to initiate the discharge.