It's the Caduceus.
Also known as the staff of Hermes, Iris and heralds/messengers in general... yes, including Mercury, who is the messenger of the Roman gods, guide of the dead and the protector of merchants, shepherds, gamblers, liars and thieves.
It is also the alchemical/occult/esoteric/whatever symbol of male and female, light and dark, good and evil and other such opposite-defined pairings, whilst at the same time representing the balance betwixt each. It also symbolises the four elements as sun, staff, snake and wing, being fire, earth, water and air respectively.
It's also used in many other contexts, most notably the Sacrifice album by Gary Numan, which is one of the many reasons I have it tattooed on my left shoulder!
However, it is NOT a proper, correct medical symbol - That is the Rod of Asclepius, which has only one snake and no wings.
Wikipedia has chapter and verse on both staffs.