Thin, even slices are key for the most tender-eating results.
Remove the bones
Stand the rested prime rib upright, holding the bones with your free hand (use a clean kitchen towel if you'd like). Using a long, thin-bladed carving knife, cut between the bones and the meat, following the curvature of the bones as closely as you can until you get to their base.
Fold bones out and remove
Once you get to the bottom, fold the bones outward as if they're on a hinge, then cut through the bottom to remove them completely.
Ready to slice
You should end up with a single solid block of bone-free meat, and a slab of bones ready to be split apart and picked at.
If you prefer, you can now remove the excess fat from the roast, leaving just an eye of meat. I prefer to leave the fat on as I carve so guests can make their own choices about whether or not to eat the fat and I can silently sort them into two categories in my mind for future judgment.
Holding the roast steady with your free hand, carefully slice the meat into thin, even slices. For best appearances, try and saw back and forth as little as possible, instead trying to cut through in a few thin, even strokes.
Don't carve it all!
Carve only as many slices as you need, leaving the rest of the roast intact. This will help keep it warm for seconds, or if you have leftovers, make it easier to store.