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.