How do you replace a ballast?
Before attempting to replace the ballast, I'd check to make sure it wasn't the bulbs first. If they're blackened at the ends, then it's the bulbs.

Otherwise, it may be the ballast.

First, cut power to the light by flipping the breaker. I'd check to make sure you got the right circuit by flipping the light switch on and off afterwards.

Next, open up the fixture and remove the bulbs. There's usually a center section of the fixture that is held in place by several flanges. Squeeze the center section to remove it.

The ballast should be inside - it looks like a rectangular box. Unscrew the wire nuts attaching it to the fixture. (I replace them on the exposed wire ends afterward just for a little extra safety.) Un-bolt the ballast from the fixture.

Take the ballast to a hardware store and get a matching replacement.

Reverse and repeat.

If you want pictures, here's a tutorial with the same basic approach.

Also, here's another resource that suggests replacing the entire fixture may actually be cheaper.