Well its not that simple, it is a word with many meanings which often leads to confusion.
Depending on your dictionary of choice, Barbecue is defined as "pieces of beef, fowl, fish, or the like, roasted over an open hearth, especially when basted in a barbecue sauce" or " to roast or broil on a rack or revolving spit over or before a source of heat". Neither of these definitions really do it justice. Traditionally barbecued meat was cooked over an open hearth, but really it is all about the smoke, which you can only get from wood, which is used as the fuel in an open hearth. The traditional definition would also be equal to grilling, but if you grill a brisket or a pork shoulder it will not turn out tender and delicious.
Anyone who has ever truly barbecued something will tell you, to get real barbecue, you need three essential elements. Smoke, low heat, and lots of time.
Now some people would argue that the heat source has to only be wood, or temperatures must never go above 250 F, but at the end of the day a lot of people can produce outstanding barbecue with multiple techniques as long as you have those three basic ingredients.
We smoke our beef and pork all night at three different temperatures to ensure moist tender meat on the inside and a nice bark on the outside every time. So come on by and see how we did.
So I would offer the following definition of barbecue. Food cooked with low heat and smoke for an extended period of time.