It's the Rod of Asclepius, not to be confused with the caduceus!

There are various theories why this symbol was invented. The one that makes most sense to me is that it depicts a (formerly widespread) parasitic nematode being removed by wrapping it around a stick to slowly draw it out of the body.

For a visual demonstration of the parasite and its removal: Bizarre Beasts - When is extinction the 'right' choice?

It was the symbol of Hermes, the Greek god of messengers. It's called a caduceus, and it was a symbol of peace that would grant safe passage to the carrier. For similar reasons it would be adopted by medical experts and come to be associated with medicine.