Lip-On Ribeye is essentially a Prime Rib roast without the bones.
The "lip" that is left on refers to a two-inch (5 cm) piece of meat that is left on at the end of the ribeye muscle. The lip piece is largely fat, with some meat in it. Because it is largely fat, the overall price per pound should be slightly lower than a cut with the lip removed (which would be just plain "Ribeye.")
This cut is 112A in the North American Institutional Meat Purchase Specifications guide.
Many restaurants serve Lip-on Ribeye instead of Prime Rib, as it cooks quickly and is easier to carve.
A Lip-On Ribeye roast can also be cut into Lip-On Ribeye steaks.
Some people feel that the lip can be trimmed off, as a great deal of it is fat and this cut of meat is already well-marbled enough. Serious beef lovers though, say to do any trimming that you want to do after cooking, as any extra fat is beneficial for the meat's moistness during cooking. They also say that they find the lip quite flavourful.
This is a well-marbled cut of beef that should be cooked by roasting (or grilling for steaks.)
Use an instant-read meat-themometer and cook to your choice of:
Medium Rare 145 F (63 C)
Medium 160 F (71 C)
Well Done 170 F (77 C)