A one-name celeb is someone who is recognizable by their first name. This does not include stage names like Sting or Madonna, only real birth names.
Examples include Elvis, Beyoncé, Oprah, Cher, Shaq, and Roseanne (at least during the 90's in the U.S., not so much now). Possibly "The Donald" counts but that may be cheating a bit by adding a word in front of it.
I think Beyoncé may be the greatest one-name celebrity.