New Orleans is the city for partying, and Bourbon Street is where everyone goes--to start off. It's full of cheap booze, cheesy clubs and bars, strip joints, drunks vomiting in the gutter, and any number of men and women flashing for beads. Cats Meow is a popular bar and has live music, sometimes good (it also has a webcam if you want to wave home to Mum). Look out for happy hour in bars like Utopia, where it's usual to get three drinks for the price of one.
But there are other places to go in the French Quarter. Pat O's is a large bar/restaurant near Preservation Hall. It serves good, reasonably priced Hurricanes (New Orleans cocktail) and has a special-effect fire fountain. Hand grenades are the other New Orleans cocktail to try--be warned, they are pretty lethal.
If you want something a bit more alternative, try Decatur Street. Walk towards the French Market end past the tacky souvenir shops and drink places for the more interesting bars. If you're into rock/alternative music, I reccomend the Whirling Dervish, a bar towards the end of Decatur Street. Another interesting club where they play anything from '80s to punk is Shim Sham, with a confusing second floor that doesn’t go anywhere.
NB: In New Orleans, you can carry your drink from bar to bar as long as it's in a carry-out cup. So buy your drink at one of the cheap "To Go" places before going into a bar. And remember to bring your ID, as you will be asked for it even in New Orleans.