New Orleans is casual during the day and casual to slightly dressy at night. A few restaurants in the French Quarter require men to wear a jacket and tie at dinner. Take comfy walking shoes.
In winter you'll want a coat or warm jacket, especially for evenings, which can be downright cold. In summer pack for hot, sticky weather, but be prepared for borderline-glacial air-conditioning, and bring an umbrella in case of sudden thunderstorms. Leave the plastic raincoats behind (they're extremely uncomfortable in high humidity). In addition, pack a sun hat and sunscreen lotion, even for strolls in the city—the sun can be fierce.
Insect repellent will come in handy if you plan to be outdoors on a swamp cruise or dining alfresco in the city; mosquitoes come out in full force after sunset in warm weather.