Grilling
Typically done at high temperatures, in excess of 400°. Meat is cooked directly above the heat source, gas flame or charcoal fire, and is cooked to serving temperature in a relatively short period of time, on average thirty minutes or so.
Indirect grilling
This is a next-step towards barbecuing. Indirect grilling occurs when the meat is cooked away from the direct heat source. For example, only one line of gas burners are lit or charcoal is heaped on one side of the grill and the meat is cooked opposite the heat source. Cooking temperatures are usually on the high side: 350° to 400° or so. Cuts of meat grilled indirectly are usually large cuts, large beef, pork roasts or whole poultry. They usually take a longer time to cook, an hour or more. Typically, indirect grilling is considered a substitution for oven-roasting meats.
BBQing
Now the definitions become a little more muddled. For my purposes, barbecue is typically done at lower temperatures than what is achieved with indirect grilling, 210° to 280° or so. The heat source is almost always away from the meats being cooked or at least the meat isn't in direct contact with the heat source. Large cuts of meat are used and are usually cooked for long periods of time, four to fourteen hours—sometimes longer. The combination of low temperature and large cuts force the longer cook times. Another identifier of BBQ is the use of wood, typically hardwood of some sort, as a heat source and/or flavoring agent. The smoke produced by the combustion of wood imparts a unique flavor to the meat. The best meats for BBQ are also unique—often they are tough cuts full of connective tissue that take a long time to tenderize. BBQ typically requires specialized equipment, ranging from expensive rigs to simple, economical backyard devices.