Can someone give me a brief, accurate run-down on the prohibition against "Passing on the Right" as it applies to the U.S.?

I’ve long believed this to be a carry-over from the days of 2 lane highways. When passing on the right meant passing on the shoulder.

I know some people believe that if you’re on the interstate and someone is driving 50 in the passing lane, that you are not supposed to drive by them on the right.

Can we clear this up?