To some extent, it depends on which part of the American South you're talking about...but let's make a generalization first.
European colonists throughout what would become the United States quickly adopted corn (Zea mays) from the Indians. They learned a technique invented in Mexico: soaking corn in alkali, which sheds the indigestible seed coat and allows the starch to form into a dough, thereby increasing the nutritional content. Corn treated like this is called hominy (see Ben Wagner's answer to What is the most important invention in human history) and is used in dishes across the United States. Common foods in the South include grits and various forms of cornbread, including small thick pancakes baked or fried on a griddle, also known as cornbread.
Pigs were probably the most common domestic animals in the early days; they could thrive on a wide variety of foods, and they were the most efficient at converting food into meat. It is said that Southerners "used every part of the pig except the scream." To preserve as much of the meat as possible after slaughter, various smoking and curing techniques are used.
So, in addition to various roasts, ribs and hams, many of the various sausage varieties are pretty traditional - you might be happier not knowing exactly how they're eaten. Common, if not unique, in the South is the skinless "breakfast sausage," usually cooked in the form of a patty. Fatty or boney parts of pigs, such as feet or backs, can be added to dishes for flavor.
"Pig's head cheese" is made by boiling pig's head until soft and sticking it together with gelatin. Even pig intestines are eaten; these are called "chitterlings" (often shortened to "chitlins"), which take hours to cook and don't smell very nice while they're cooking. Pork intestines are associated with African-American cooking—dating back to slave times, when slave owners kept the best meat for themselves and gave the less tasty portions to their slaves. I can find them in my local supermarket today, but I have never eaten them and don't know anyone who has. Of course, here in the South we also developed a technique for tenderizing tougher cuts of meat by cooking it extremely slowly over a smoky wood fire - I'm talking, of course, about grilling. (Sorry, people cooking hot dogs and burgers on their backyard grills are enjoyable and fun, but that's not "barbecue.") Barbecue styles vary across the South. In Texas, beef and sausage are preferred; in North Carolina, pork shoulder is popular; in Memphis, pork ribs are famous; and in parts of Kentucky, lamb is a favorite. North Carolina uses a vinegar-based finishing sauce, South Carolina likes a mustard sauce, Memphis favors a dry rub (a dry mixture rubbed on the meat for seasoning), and Kansas City likes a sweet ketchup-and-molasses sauce, which I like best in The Arkansas joint offers a list of six different sauces for those who like to experiment.