My 2004 Chevolet Cavalier was running and started yesterday, but the next day would not start. It would turn over, but not start or run. Just run my battery down, started over, over

SOURCE: car won't start. turns over but stalls.

sounds like a fault in the theft deterrent
is the theft light on or flashing
you will need to have the scanned for codes

SOURCE: My battery keeps dying but my alternator is fine

First, DON"T check your alternator that way anymore. It's not good to do on newer systems! The best, quickest check I can tell you is to put a volt meter across the battery. Running with accessories off you should see voltage of about 13.5 to 14.5.
With all accessories and headlamps on high turned on, it should not drop below 12.5. (and that is the absolute low end!).
It sounds like something in the car is staying on. check the glovebox light and interior light delay and make sure they are OK. Otherwise, you will have to start checking entire system for a draw that's taking out the battery overnight. To check, you can pull one fuse at a time and see if draw goes away, then check system that fuse serves.