The New Orleans Convention and Visitor's Bureau has put together a list of recommended places to eat here: http://www.neworleanscvb.com/nola-micro/restaurants/
For my personal recommendations, I would recommend:
1. Café Du Monde for Beignets (a square piece of dough, fried and covered with powdered sugar) http://cafedumonde.com/
2. Jacques Imos Café It is extremely eccentric. So be prepared! Shrimp to Alligator is on the menu(the decor is funky and you might even get to eat in the pickup truck parked in front of the café). Book in advance. http://jacques-imos.com/
3. NOLA Restaurant by Emeril It is Casual, Funky and in French Quarter. Southern Cajun, Vietnamese and Southwestern influences using Louisiana products. http://www.emerilsrestaurants.com/nola-restaurant
4. Commander's Palace It is expensive, so this is a recommendation if only you have budget for it and want a special experience. It is in the Garden district and I believe a jacket is required for the dinner time. No shorts or t-shirts, or open-toed shoes for men. You can take the St. Charles Street Car to get there (I did and it was a really nice way to get there and back). It is fine dining and maybe something to consider if you are coming in early and want to do Sunday brunch! www.commanderspalace.com
5. Pat O'Briens for a Hurricane. It has a piano lounge and the show starts at 6 PM. It gets packed when an event is in town, but it is worth the trip just to try the authentic hurricane. http://www.patobriens.com/patobriens/
Enjoy! Let me know if you make it to any of these places and tell me what you thought!