BBQ is actually a hotly debated topic in the states, no pun intended. In several parts, BBQ refers to any outdoor grilling event, in which case you might want to know about all the different things that can be cooked over coals. These should be called cook-outs. There's little science behind grilling beef patties and sausages, it's easy enough for college graduates to understand. I'm sure Brits enjoy this as much as Americans do. However, in the heart of BBQ culture, BBQ is nothing like cook-outs. I'll assume you refer to the refined, Southern technique of slowcooking smoked meats.
First the hardware: If you have a lot of money, the best device for the job is a Traeger grill. However, there are ways of fashioning smokers out of basically any container (I'll echo the recommendation for Alton Brown's "Grilling Essentials" linked by @CosCallis). The key is being able to maintain a constant, low temperature for a very long period of time.
Then the software: A good BBQ sauce serves as a marinade, a basting solution, and a finishing sauce. These always have robust smokey, salty, sweet, sour, savory, and spicy notes, adjusted to the preference of the griller. Adobo, mesquite, worcestershire sauce, ketchup, apricot jelly, apple cider vinegar, soy sauce, tabasco sauce, honey, beer, and bourbon are all really common ingredients that are concocted for a homemade BBQ sauce.
Another important aspect of your software is wood for smoking. Applewood is probably the most popular smoking chip for BBQ, followed by mesquite and cedar. The wood adds a tremendous amount of flavor and aroma during the smoking process.
Temperature is really important. The philosophy of "low and slow" is essential to BBQ. Beef and pork can be smoked at 140*F (60*C) for as long as 18 hours to melt connective tissue and give a thick pink smoke ring in the meat as deep as 1" (a couple cm) the deepness of this pink ring is a crowning achievement of good BBQ. Poultry needs to be smoked at higher temps to kill salmonella. Fish can be cold smoked.