Definitely second the opinions above on Emeril's for lunch. Do try the banana cream pie!
I'm surprised no one's mentioned Commander's Palace yet. I absolutely enjoyed my dinner there, and the bananas foster souffle will live on in my dreams for many years.
If you've never tried Cafe du Monde, eh.. it's not that great, but worth trying once just for the experience.
Don't be afraid to just walk into places and try them. I was amazed at the quality of food in this great city.