A nanny is similar to a live-in servant. They tend to be older ladies who have some training and are quite expensive; only a rich family would have a nanny. Au pairs are foreign students that, as Antonio said, live in the house in exchange for baby-sitting service. The services should not be an all-day thing. If you, or someone you know, is thinking about being an Au Pair make sure that everything is agreed to prior to the trip. Au Pairs are unfortunately taken advantage of; it doesn't matter the country. For instance, an Au Pair should never have to do housework beyond simple childcare related duties. Babysitters are usually local teenagers that will watch the kids for the evening.

HI Julia! I guess babysitter is somebody who looks after someone's children for a couple of hours whereas an au pair is a kind of exchange. You go to a foreign country, live with a family without paying any rent, but you have to take care of the children when needed (like for example getting them to school).