A bone-in ribeye, aka a standing rib roast, is a cut from the back of the steer's upper ribs, an area that can have the most marbling in the animal and often the most flavor. If you still want that flavor but need the bones for something else -- stock or deviled beef bones, for example -- remove them before you put the roast in the oven. It's much easier taking out the rib bones when the meat is cool than when it's hot and your guests are hungry.
You should use a sharp butcher knife with a straight blade that's at least half the length of the rib bones. Anything less than half will make separating the ribs from the roast much more difficult. You'll also want to make sure the handle of the knife is firmly attached to the blade before starting; this is a large cut of beef that will take some elbow grease to butcher.
Begin with the rib side up and the ends of the bones pointing to the side of your dominant hand. Insert the knife under the top rib and cut down, underneath every bone. When you get to the bottom, slide the knife horizontally under the entire bottom rib. Pull the ribs away from the meat and then cut them off by sliding your knife in the opposite direction.
Use a steady, firm knife pressure as you move the blade under and around the rib bones. A jerky or start-and-stop motion won't provide the smooth, even cut needed to detach the bone from the meat. A sharp knife is the key to ensuring your knife responds to the pressure you're using.
As when butchering any piece of meat, there is a risk of cutting yourself when removing the bones from a ribeye. Pay special attention when sliding the knife horizontally under the bottom rib; this is where the knife is most likely to slip off the meat and slice your non-dominant hand. When pulling the ribs away from the meat, you may need to use the knife to cut some of the connective tissue away. Use short, small cuts to reduce the risk of injuries to your arms and torso.