You have to go to these restaurants:

• Verti Marte: for your late night delivery needs: anything from a pack of smokes, bottle of a bourbon--or a great Muffaletta sandwhich 
  
• Willie Mae's Scotch House: Far and away the best fried chicken -- anywhere

• Cochon: Donald Link's terrifically pork-centric restaurant

• Domilise's Po-Boy & Bar: for arguably the best Po' Boys in town

• Restaurant August: possibly the best example of NOLA fine dining

• For some tunes, be sure to catch Kermit Ruffins live at Vaughan's Lounge.

There are quite a few!

• Shrimp and grits -- the best place that I experienced this was at Surrey's Juice Bar.  Here's my list of shrimp and grit spots in NOLA.
• Muffuletta -- Central Grocery and Cochon Butcherare the places to get these
• Po 'Boys -- Domalise for sure
• Beignets and cafe au lait -- I know it's two things, but they go hand in hand!  Café du Monde and Morning Call are the best.
• Fried Chicken -- Willie Mae's Scotch House
• Charbroiled/grilled oysters -- best discovery I made in NOLA!  Acme Oyster House and Casamento's are the places to be.
• Snowball -- Hansen's Sno-Bliz and Pandora's SnoBalls (keep in mind that this is seasonal and available during hot weather by NOLA standards)
• Southern cocktails -- look for mint juleps and sazaracs. Any nice restaurant will likely have a good menu.

You can't just eat the entire time, so I also suggest that you hit the Southern Food & Beverage Museum, which also contains the Southern Cocktail Museum, when you're taking a break from eating.  This will for sure make your trip a very food-themed one.

Might want to fast for a week before you head to NOLA ;)

It is hard to add to these amazing answers, but I'm going to add the following:

• Ronnie's Boudin & Cracklin House in Baton Rouge. ( (http://ronniesboudin.com/)

  (only if you're driving to NOLA) For those who aren't in the know, Boudin is sausage made from ground up pork, onions, peppers, celery, rice, and lots of spices. Ronnie smokes it. Generally when you get Boudin, you're actually getting Boudin Blanc (non-blanc has blood in it, which I am less keen on).

  If you have any sense, make a stop and grab some hot boudin or boudin balls (a fried ball of the stuff). You can also get some packages by the pound and it freezes up well.

• Cake Cafe & Bakery - Great breakfasts.  Shimp and grits is fabulous as well as boudin and eggs.  Everything comes with an amazing biscuit.
  
• Killer Po-boys - this is an amazing Po Boy place hidden in the back of the Erin Rose Bar.  They serve somewhat less traditional po boy recipes but they are CRAZY good.
  
• Commander's Palace - Won best executive chef (James Beard) in 2013.  Amazing weekend brunch with great lunches and dinners.  Lunch is the cheaper option and more casual.  Dinner tends to be more expensive and the expect you to dress up a touch.  No tshirts no shorts for any meal.
  
• Cure - An amazing cocktail bar that takes cocktails VERY seriously.
  
• Three Muses - Amazing gastropub/venue on Frenchman.  Music every night.  Outstanding cocktails.
  
• Jacques-Imo's - This is a drive out of the French Quarter but it's worth it.  Still spendy, but casual/loud/fun.  No reservations and expect a long wait.  Get the alligator sausage cheesecake.
  

Definitely try the beignets at Café du Monde!!!

Everyone did a great job covering all of the must-eats! However, I'm surprised no one brought up the infamous southern PRALINES!!!!! These buttery, sweet, melt-in-your mouth candies are a specialty in New Orleans. When you eat butter pecan or pralines-n-caramel flavored ice creams, these are the little candies that inspire them! The top-notch spot to get these is  Southern Candymakers, right along the border of the French Quarter. It's a really easy-to-miss little shop right next to the historic  Bienville House Hotel. You'll see an endless trail of chain candy shops advertising 'Best Pralines in New Orleans' along the tourist-focused strip of shops along Decatur Street- while these places offer endless free samples, I must warn you to not make your purchases until you've made it to Southern Candymakers. They are the best. 

When I was in NOLA last week, I also discovered a really quaint little bakery that had all kinds of tempting-looking candies, Bittersweet Confections! It was on Magazine Street on the way to the  The National WWII Museum, which is definitely worth a visit for something a little different than what the rest of the city has to offer. 

Also, Italian cooking isn't something you'd expect to be great in the city famous for Mardi Gras, but there is one place I must pay tribute to-  Irene's Cuisine. If you've had enough of the cajun flavors but still want a great, homecooked meal in a hearty, family-style atmosphere that you can picture thriving back in the days of black-and-white nickelodean films, this is the spot. It's been there for years, and I can tell why after eating their oven-roasted chicken. They still hold a dessert menu that's true to the area's southern roots, with a GREAT Creole Cream Cheesecake. 

Have fun and be merry!

Other than  Café du Monde for beignets and hot chocolate? Everyone talks about  Mother's Restaurant for poboys..if you are wanting something a bit nicer? The Italian restaurant Vincent's Italian Cuisine on St. Charles street is always good. Sylvester Stallone's favorite when he is around. A little heavier but most food there is! :)

Shrimp & Grits (and 25cent martinis) at Commander's Palace Restaurant

Beignets and chicory coffee at Café du Monde

Wood-fired oysters at Cochon Restaurant

Muffuletta sandwich Central Grocery

Fried Chicken at Willie Mae's Scotch House or Dooky Chase's Restaurant, Sazerac at Carousel Bar....enjoy!

Depends on season:

- Oysters (seasonal- jan/feb-early summer)

- Po-boy (just a hoogie, but get it w fried fish- shrimp i like the best, but can be anything. Many people like to get roast beef, but other cities to a "french dip" better.)

- Crawfish (same as oysters)

- Jambalaya

- Red Beans and Rice w Sausage

cajun dinner

Gumbo Shop (http://www.gumboshop.com/)

Jacques-Imo's (http://jacques-imos.com/)

Soul Food

Willie Mae's Scotch House

http://www.thrillist.com/venues/willie-maes-scotch-house