An actual Dremel is a brand name, power, rotary tool with a variety of attachments that let you grind, sand, polish etc. You can get one at most hardware stores. With that said, like Kleenex and tissue, the Dremel brand name is now used by some folks to describe any similar device.
For our purposes here, it can either be the actual Dremel tool (which some people get because it can be used for projects other than grinding pet nails) or any of the pet specific rotary grinders (which, according to the kind people at Petco, tend to be cheaper and are designed for less noise and vibration). The last time I was evaluating the pet-specific tools, I found some that had an actual grinding stone and some that had emory-board or sand paper glued to a rotating drum.
Savannah needed to get used to nail clipping first, so we still use nail clippers, but there is a dremel or dremel want-to-be in my near future.