Here's what happens if certain parts are bad on your fluorescent fixture:
1) Ballast good, Starter good, bulb bad - usually you'll see that the end of the bulb are darkened. Sometimes the ends will light up partially, the bulb may try to light but look dim, or it will be completely dead. Easy to fix - just put in a new bulb.
2) Ballast good, bulb good, starter bad - If you've put in a new bulb and the bulb does nothing, lights on the ends but won't light fully, flickers on and off quickly, or takes more than 15 seconds to light, then your starter is bad. Sometimes you can get the bulb to light by removing the starter while power is on, but it won't light again after you remove and reapply power. Easy to fix - put in a new starter.
3) Bulb good, starter good, ballast bad - this is a little trickier, because if the ballast is bad, then your starter and bulb are probably bad, too. If you put in a new starter and a new bulb, power it up, and the bulb flashes quickly and then stays out, then your ballast is bad and just shorted out your new bulb. New ballasts can be bought for about $5-6 at Lowes or Home Depot. Just get the ones rated for 25W or less. I usually keep bulbs around that are semi-bad - meaning they are dark on the ends or only light at the ends. If I have a game with a dead bulb, I stick the semi-bad bulb in first to see if it blows immediately - indicating a bad ballast. This keeps me from blowing new (and expensive) bulbs. I remember Ken Layton once mentioning a way to test a ballast with a meter, but I've forgotten what he said. Perhaps I'll look for the info later....