The ballast is a black-box that is about 2"x4"x8" that does some physics voodoo to heat-up the filaments in the fluorescent bulb so that electrons are boiled off, as well as acts as a voltage regulator to keep the lamp from blowing the circuit. The one at home went dead last week while I was away (according to mom, pretty much as soon as I put my foot out the door; the appliances are all evil machinations that take advantage of poor old ladies unskilled in the arts of intimidating them). Took me some time to change the ballast, and since the instructions are not readily available on Google (the correct instruction only appeared on the 3rd or 4th page), I thought I'd increase the number of pages that gives a set of instructions. Notice that when you go buy the ballast from places like HomeDepot, the boxed items usually also contain a detailed, illustrated set of instructions.

The ballast is not small, white, and round. That is the starter. Google can help one find easily how to change a starter (thought many of those pages mis-identify themselves as instructions to changing the ballast instead).

1. The most important step!. Turn off the power at the light switch and on the circuit breaker. It would not be fun to touch a 120V hot wire.
2. Remove the fluorescent bulbs: turn the bulb a quarter turn or so while pulling gently on it.
3. Remove the metal reflector that also hides all the wiring and the ballast.
4. Follow the wires from the ballast, the number of wires depend on the type of ballast. Mine is a class P that serves 2 F40 style lamps, which gives is 8 wires total. If you see wire nuts, remove them. If you don't see a wire nut on a particular wire, cut it about 3 or 4 inches from the ballast.
5. After all the wires are cut or the wire nuts undone (so the ballast is not connected to the fixture by any wires), remove the ballast by unscrewing the one nut or two that holds the ballast to the fixture. The ballast is quite heavy, so make sure you don't drop it on your own head.
6. Now, inspect the loose wires. Usually new replacement ballasts bought in a store will come with the wires pre-stripped. Make sure the ends are not damaged. For the wires on the fixture, if
   • they were unwound from the wire nut, it is best to just nip off the parts that were in the wire nut, straighten the line out, and strip off a 1/2 inch.
   • they were cut, then just strip off a half inch.
7. Put the new ballast in, the same way the old ballast came out (i.e., with the nut).
8. Here's the tricky part: read the circuit diagrams printed on the back of the two ballasts, and see if they are identical. If you get two of the exact same model and manufacturer, then you should be set. If not, take care to see that the colors match up on the wires. Beside the government regulated rule that black and white must run to the power lines, the other colors are not required to be the same, especially on older models. Follow the diagram to pair up the wires. (More likely than not, though, the colors will match up: red to red, yellow to yellow, blue to blue.)
9. Hold the two stripped ends, one from the ballast, one from the fixture, of the pair parallel to each other, insert them into a new wire nut, and twist. Then wrap the nut, any exposed wires, and the entire connection in electrician tape.
10. Replace the reflector, take care not to pinch any wires.
11. Replace the bulbs.
12. Turn on the circuit breaker, and turn on the wall switch.