I am very unknowledgeable when it comes to cars. My mom owns a 1998 Toyota Camry. Recently the A/C stopped working and she took into a mechanic. They indicated that her compressor was shot but one mechanic said she could still use the defroster. However, we had a nasty rainstorm and when she turned the switch to use the defroster, the fuse blew and nothing. Is it true that you can still run a defroster without the compressor?

Yes, you can still use the defroster, but no dehumidified air will come out, just plain air. Might make things worse on humid days.
Sounds like the A/C fuse took a hit and caused things to stop. The compressor clutch has a coil in it with copper windings. Once they melt, a short has occurred and pops the fuse.
You could try replacing the a/c fuse, then disconnect the big connector at the compressor till it's fixed.