The word "prime" is a legal designation that refers to the fact that the beef has been graded prime by the USDA. Restaurants that serve prime rib must use prime beef. Otherwise, they're required to call it a standing rib roast, which is decidedly less exciting sounding, or simply a rib-eye roast for the boneless version.
The primary characteristic USDA inspectors look for when assigning grades is the amount of fat within the edible sections of the meat. This so-called intramuscular fat, or marbling, is a source of flavor and moisture. Thus the more marbling there is, the higher the grade the meat will receive.
The rules governing meat inspection state that the degree of marbling, and thus the grade, of each side of beef, shall be determined by examining the ribeye muscle (the longissimus dorsi) between the 12th and 13th ribs. The other main factor is the age of the animal. To be considered prime, beef cattle must be 9 to 30 months of age, although most are younger than 24 months.
The reason age matters is because age is related to tenderness. Younger animals will provide more tender meat. Inspectors can determine the age of the animal by examining the chine bone (aka the backbone). In younger animals, there is a small bead of cartilage where each rib meets the chine bone. In older animals, this cartilage turns to bone.
These criteria mean that prime beef will be tender, moist, and flavorful. Having said that, the rules regarding calling something "prime rib" only cover restaurants and butcher shops. If you prepare a beef rib roast home, whether it's boneless or bone-in, you can call it anything you want.
For that matter, since retail meat isn't always graded, you can certainly find some very high-quality beef that doesn't happen to have the prime designation. Simply look for the marbling.
Preparing Prime Rib
In general, they involve applying a high amount of heat for a short time in order to produce a flavorful brown crust on the exterior, then roasting at a lower temperature for the remainder of the time.
Usually, the high-heat stage comes at the beginning, but it's possible to slow-roast a prime rib and finish it with a high-temperature sear right at the end.
Not all prime rib is prepared bone-in, but the bone adds flavor and moisture. Also, the rib bones make it easier to roast the prime rib since they act as a natural roasting rack. Even so, when cooked to perfection, a boneless prime rib can be every bit as sublime as the bone-in kind.
For a classic prime rib recipe: Prime Rib Roast: Traditional Method.