You don't have to wear a costume if you don't enjoy that kind of thing. However, when you look around, you may feel stranger out of costume than in! (You might want to at least don one of the artistically diverse masks sold on every street corner.) Costuming is big business in New Orleans.
Originally, costumes were worn to keep the identities of krewe members secret. Today, the secretiveness is no longer a big deal. However, you can still risk your membership in older krewes, like Comus and Rex, if you take off your mask during the parade, though even they usually forgive krewe members for moving their masks slightly to drink more easily or to kiss a happy bead recipient!
Veteran parade-goers warn newcomers to dress comfortably -- either in costume or streetwear -- for the parades, to bring a bag for their throws and extra tissue for use in the port-a-lets scattered around the city. You can check parade routing and view the spectacles from curbside, or you may want to enjoy Mardi Gras by purchasing tickets to the city's reserved grandstand seats from Ticket Master.
Also, if you're going to New Orleans during Mardi Gras, make sure your trip is a safe and pleasant one by checking out the survival tips offered by the official Mardi Gras site.