The choke has two main functions. It (in conjunction with the starter if it has one) causes the tube to ignite by using the back emf to create a plasma in the tube and it controls the current through the tube when it is ignited.
In a gas discharge, such as a fluorescent lamp, current causes resistance to decrease. This is because as more electrons and ions flow through a particular area, they bump into more atoms, which frees up electrons, creating more charged particles. In this way, current will climb on its own in a gas discharge, as long as there is adequate voltage (and household AC current has a lot of voltage). If the current in a fluorescent light isn't controlled, it can blow out the various electrical components.
A fluorescent lamp's ballast works to control this. The simplest sort of ballast, generally referred to as a magnetic ballast, works something like an inductor. A basic inductor consists of a coil of wire in a circuit, which may be wound around a piece of metal. If you've read How Electromagnets Work, you know that when you send electrical current through a wire, it generates a magnetic field. Positioning the wire in concentric loops amplifies this field.
This sort of field affects not only objects around the loop, but also the loop itself. Increasing the current in the loop increases the magnetic field, which applies a voltage opposite the flow of current in the wire. In short, a coiled length of wire in a circuit (an inductor) opposes change in the current flowing through it. The transformer elements in a magnetic ballast use this principle to regulate the current in a fluorescent lamp.
A ballast can only slow down changes in current -- it can't stop them. But the alternating current powering a fluorescent light is constantly reversing itself, so the ballast only has to inhibit increasing current in a particular direction for a short amount of time.
Magnetic ballasts modulate electrical current at a relatively low cycle rate, which can cause a noticeable flicker. Magnetic ballasts may also vibrate at a low frequency. This is the source of the audible humming sound people associate with fluorescent lamps.
Modern ballast designs use advanced electronics to more precisely regulate the current flowing through the electrical circuit. Since they use a higher cycle rate, you don't generally notice a flicker or humming noise coming from an electronic ballast. Different lamps require specialized ballasts designed to maintain the specific voltage and current levels needed for varying tube designs.
Fluorescent lamps come in all shapes and sizes, but they all work on the same basic principle: An electric current stimulates mercury atoms, which causes them to release ultraviolet photons. These photons in turn stimulate a phosphor, which emits visible light photons. At the most basic level, that's all there is to it!