Broiling is a high heat cooking method in which the heat source is above the food.
Grilling would have the food on a metal lattice and the heat source under the food.
Bar-B-Que involves a sauce and slow cooking; but, that term can cause some arguments among certain groups. I am from Texas, and am used to slow cooked beef brisket with a tomato based sauce. North Carolina is used to slow cooked pork and a mustard based sauce. That just scratches the surface. There are some pretty good articles about the subject on Hubpages.
FYI - A great deal of the american cooking terms are actually French or have their roots in French cooking. This is especially true in the southern states like Alabama, Louisiana, and Mississippi.