Restaurants that serve prime rib are required to use beef that has been graded prime by the USDA. Otherwise they have to call it a standing rib roast, which is decidedly less exciting sounding, or a rib-eye roast for the boneless version. Generally speaking, meat that's graded prime is tender and has a generous amount of marbling. But if you make prime rib at home you can call it anything you want. And 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.
The traditional way of serving prime rib is to roast it medium rare, and there are a number of techniques to accomplish this. 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.
Also see: How to Roast Prime Rib
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 very classic prime rib recipe see: Prime Rib Roast: Traditional Method.