The French Quarter has a range of shops, from little individual one-off stores to the usual big department names.
The shops in the main streets vary from tourist tack to art galleries, mostly depending on location. Royal Street has mainly upmarket shops--expensive (but lovely) dress stores, carefully crafted masks, beautiful jewelry stores, Civil War armaments.
At the other end of the scale is Decatur Street, with the usual souvenir shops of beads, T-shirts, and voodoo dolls. There's also a small shopping mall with a Virgin Megastore attached. (The food court upstairs isn't that great, but you can eat your snack on the outside balcony by the Mississippi River.) Walk on down Decatur Street and you'll come to the French Market, a good place to shop for cheap souvenirs. The shops at this end are more alternative, with some great clothes shops if you're looking for something different. Gargoyles has some fantastic outfits and has three outlets selling clothes, shoes, and leather.
Check out the side streets in the Quarter as well for a range of shops from Harley Davidson to authentic voodoo charms (allegedly) to one-of-a-kind corsetry. If you long for the usual brand-name retailers instead, wander down to the Riverside shopping mall near the aquarium. It has all the regular names--Gap, Abercrombie and Fitch, etc., but with the added bonus, this being New Orleans, of being able to get a cocktail in the food court.
There's another shopping mall, more expensive, near the beginning of Decatur. The main store is Saks and the other shops are correspondingly more upmarket, with a small cinema attached on an upper level.