Thanks for all the replies. I chose SHA1 for the time being, and it works okay so far. I had to do some research on implementing it as I haven't done it before.
MarkR had a good point about database security. BUT - nothing is invulnerable.
I'm pretty certain that what I'm doing now will suffice, with one exception. I haven't quite figured out how to do a "three-strikes-you're-out" login page.