Screams New Orleans:
Commanders Palace: doubles as one of the best brunch places
Galatoire's (Friday lunch institution, if you'll be there then, but anytime)
Peche: order the whole roasted fish! Maybe the best dish in town.
Oysters
Luke: happy hour
Casamento's: super old-school tile floors and walls (yes, they hose it down)
Brunch:
Brennans
Atchafalaya
Anytime awesome:
La Petite Grocery
Coquette
Herbsaint
Couchon (I'm more into fish places, but you gotta give this meat props!)
Brigsten's (old school)
Dickie Brennan's steakhouse
Mr B's (someone above didn't like it, but I'll stand by the fried oysters, gumbo, bbq shrimp)
Po Boys:
Domilise's (for extra credit: Peyton's favorite HS hangout)
Actually, everyone's recommendations on this thread are good and I'm getting hungry
Make reservations. You can always change or cancel them, but a lot of these places fill up. It's not a last minute drop-in food town. You can even make reservations at two different places at slightly different times and decide later, but don't be a d**k and fail to cancel because somebody is trying to get in