The top end of Duval Street, near Mallory Square, is where most of the bars are. Key West tries its best to be a party town reminiscent of New Orleans, with bars opening early and closing late--it even has a couple of New Orleans-themed bars. Of course, this does mean that there are lots of very drunk and mostly middle-aged tourists in the bars along the street.
Margaritas are the classic Key West drink and the Margaritaville Bar specializes in them. Sloppy Joes is a main tourist hangout, with a large and noisy bar with regular live bands and a dance floor. I'd reccommend checking out what kind of music is playing before you go; one night they had a really energetic cover band there that did a good mix of modern hits and oldies--I had a blast. We went back the next night and they had a comedy band there, which involved a lot of swearing. They ended their act by persuading several girls to come up onstage and flash the room--one girl was particularly impressive, as she had superior muscle control and could actually make her breasts move independently.
If you prefer something slightly less obvious, try the bars down by the waterfront where you'll get more of a local and tourist mix. The Blue Parrot is a popular and pretty cool place to have a drink. Alternatively there are a few restaurants along Duval Street that open fairly late and have courtyard dining where you can drink in comfort and watch the (Key West) world go by.