This subject is near and dear to my heart. My absolute favorite way to celebrate a special occasion- a birthday, a holiday, good news, etc.... is to get dressed up and go out to an upscale restaurant for a great dinner. I've been to a lot of great restaurants in New Orleans, but some really stand out.
Restaurant August- Chef John Besh is, in my opinion, one of the best in New Orleans and August is a wonderful restaurant. I ate at August for my 26th birthday and I still remember the wonderful duck with duck confit, foie gras, and tempura cherries that I had as an entree. The service was wonderful, and they actually mailed me a thank you card for coming to their restaurant. (I also love Besh's newer restaurant, Luke, for a slightly more casual experience. And if you go there, do try combining an order of the truffled pate with a dozen fried oysters. It sounds strange, but you won't be sorry! It is decadent!)
Lilette- I actually took my dear boyfriend here for his 27th birthday, which we celebrated on the same day that I defended my Master's thesis. I was amazed by the yellowfin tuna, which was served with beets, ginger, basil and mint. The combination looks strange in print, but the taste was amazing!
Bistro Daisy- Bistro Daisy is fairly new- my boyfriend and I celebrated out anniversary there this year. Bistro Daisy combines great food with a lovely, intimate environment and wonderful service. I particularly remember my excellent appetizer (the special)- foie gras with rhubarb compote. It was just wonderful.
Cochon- Cochon is more casual than the places listed above, but Donald Link is another of my favorite local chefs and so far I've loved all of the restaurants he's had a hand in (Herbsaint, Cochon, and Cochon Butcher/sWine Bar- I have yet to try his catering at Calcasieu). Cochon shines because of its menu- an upscale twist on traditional southern food. And Cochon's cornmeal pineapple upside down cake is my all time favorite dessert.