French Quarter
You'll be staying there, so you'll necessarily see it. Rather than a place with a single character, the Quarter was until about 1800 almost the entire city, and therefore needed to serve every human impulse. Therefore, the filth and debauchery of Bourbon Street coexists almost side by side with the elegance of famous restaurants (some of which date back to the 19th century), antique shops, museums and book stores. It bears emphasizing that New Orleans drinks, particularly on lower Decatur St. (Molly's, Turtle Bay, Mojo, Spitfire, Tiki's, The Abbey) tend to be much stronger and less expensive than those you'll find out of town. The Quarter still retains a bit of the aspect of a self-contained city with grocery stores, a school, a walled park, a convent (in use until quite recently) and what amount to several distinct entertainment districts. Locals who don't work there rarely go to Bourbon above St. Ann St., for instance, but the oldest bar in the United States (Boston disputes this but not with much energy) is Lafitte's Blacksmith Shop at the relatively quiet corner of Bourbon and St. Philip, in a building dating from somewhere between the 1720s and 1770s, depending on whom you believe. Lower Decatur (downtown * of St. Philip), is where many locals feel almost as much at home as in their own houses. The Quarter also contains the infamous Lalaurie Mansion (which is privately owned - the exteriors in American Horror Story were actually of the Gallier House, a museum two doors down), the Voodoo Museum, Wax Museum and Pharmacy Museum, in addition to the places where you'll want to buy clothes, shoes (including a Fluevog store soon), makeup, wigs, and, if you really must, a thousand varieties of hot sauce, t-shirts and back scratchers made from baby alligator heads. The French Market is a shaded but open-air market existing since about the 1830s in the lower Quarter near the river. It was formerly the site of Gallatin St., known in its day as one of the most dangerous places in the world, which was therefore torn down. Postcard-style pictures of the Quarter are easy to find, so for how locals view the place, we recommend you look here instead.
* "Downtown" is usually used to indicate a direction rather than a location in New Orleans. Hence, what most cities call "downtown," we call the CBD or Central Business District. This is probably due to the fact that the points of the compass are useless in giving directions here (the sun actually rises over the West Bank of the Mississippi as seen from some parts of the city (when it's said to set over the W'ank in A Confederacy of Dunces, this is actually a local in-joke), so we navigate by whether things are upriver ("uptown" - which also designates a location), downriver ("downtown"), or more towards Lake Pontchartrain or the river ("lakeside" and "riverside").