Although you'll often hear the two terms used interchangeably, grilling and barbecuing are two entirely different forms of cooking. Typically, people who say that they are "barbecuing" mean that they are grilling, they're just misinformed. Grilling involves cooking directly over an open flame, whereas barbecuing typically involves meat being smoked for an extended period of time.
Grilling is a less time intensive method of food preparation than barbecuing. You take your meat or vegetables, season them a bit, and then place them on your grill directly over the flame. Cooking times can vary from just a few minutes for a hamburger or hot dog to twenty or thirty minutes for chicken or steak.
A grill may be charcoal or gas, but despite what others might say, the taste difference is slight. Essentially, It all comes down to convenience. A gas grill can be lit and ready to go in a matter of minutes, whereas a charcoal grill takes a lot more time to prepare for cooking. Most people agree that food cooked on charcoal grills tastes better, but unless you're grilling for a larger group of people, the time factor just isn't worth it.
Barbecuing, on the other hand, typically involves a smoker. A dry rub or barbecue sauce is applied to the meat to be cooked and it's placed inside a smoker to be cooked. Cooking times can vary from an hour to several hours, maybe even half a day. Barbecuing consists of slowly cooking a meat, so that the smoke from the wood you're using can be absorbed into the meat to add flavor.
There are a wide variety of ways to accentuate the barbecuing process. People may use hickory or apple woods to infuse flavor into the meat. There are countless rubs and sauces you can apply to the meat before and during the cooking process. Some people even put beer or other liquids in the smoker to add steam to the process. Barbecuing is a far more diverse cooking method than grilling, but one thing holds true: The quality of barbecue is derived from slowly cooking the meat to maximize the flavor.
Even when a person barbecues chicken on a grill, he is not really barbecuing, but grilling. Rather than barbecuing, we should probably use the term "smoking," and reserve grilling for anything cooked on a grill in a relatively brief amount of time.