Barbecuing is cooking food with dry heat and hardwood smoke. Spices, a marinade, or sauce may be applied to the food as part of the process. Any heat source can be used, but grilling as a barbecue method is hotly debated in most of the US. In places like eastern N.C. and Texas, they believe grilling is not a barbecue method. Using hardwood will infuse foods with a rich smoky flavor. Different woods will affect the flavor and cooking time. What woods to use is completely up to you as long as its hardwood. Charcoal can provide more predictable cooking times, the smoke flavor, but without the unique flavors that wood offers. Gas and electricity can allow more temperature control than charcoal and wood, but like charcoal doesn't provide the rich flavor that wood offers. What method and heat source you use is completely up to you. Many people also argue about what meat to use. In eastern N.C., they consider only the whole hog to be barbecue, while in Texas they only consider beef to be barbecue (Sorry about the repeated references to N.C. and Texas.). However, for the common American, barbecue means pork.
Many commercial BBQ sauces are actually sugar-based, not tomato based. Therefore, I don't advise coating your food in commercial based sauce right before you cook it unless you want food burnt on the outside, but raw on the inside.
For more information, see Barbecue Tips.