"Au Pair" means "on par" meaning living as part of the family. As such, au pairs get house and board and a small amount of pocket money, but no salary as such. They also don't have clear-cut work duties, they help around a house, as a member of the family would be expected to do, cooking, cleaning, washing etc. As a parent, you're generally not supposed to leave the au pair in sole charge of the children for more than brief periods of time (as you would a babysitter). That's the general theory, of course, individual arrangements range widely.

A nanny is a professional employee of the family, generally has training and certification, has clear duties and makes an actual salary. An "au pair nanny" is an oxymoron, that would be a "live in nanny" and, yes, definitely rich people stuff. Au pairs are definitely possibilities for the middle class.