What is the symbol for the RN nurse, the wings and snakes, called?

Patricia is somewhat correct.

The winged symbol is entitled a caduceus. The rod with a single snake is the Rod of Asclepius. Technically they are not interchangable. The caduceus was originally a symbol of commerce and some physician's groups use the Rod and not the caduceus.

Another medical symbol that has a snake is the symbol for pharmacy, the Bowl of Hygieia.

caduceus

It's called a caduceus and it is the symbol of medicine, not just nursing. There are two versions, one with or without wings and two snakes and another with a single snake and shaft. It's also called the Rod of Asclepius, and was used as far back as 2600BC. It is associated with priests in ancient Greece, who sometimes used snakes in healing rituals in their temples, and is also a symbol of a god of healing.

The symbol that is actually primarily attached to the RN is the oil lamp which is symbolic of Florence Nightingale, considered the founder of modern nursing. One of her nicknames was "the lady of the lamp."