What is the medical symbol called? What does it mean?

I believe you're referring to the image on the right...

It's called the Staff of Asclepius.

The story? Asclepius was apparently a highly successful Greek healer of many hundred years B.C.. Several hundred years later, he was actually deified, and became worshiped as none less than the god of medicine, and the son of Apollo. (Nowdays, we doctors have to rest contented as mere pillars of the community.)

But why this distinctly serpentine symbol for him? This matter is a little controversial, but most authorities believe that the snake represented rejuvenation to the ancient Greeks, since it sheds its skin. The rod is simply the common symbol of authority.

Aesclepius naturally had offspring, two of whom are famous in Greek mythological circles - Hygiea (Health) and Panacea (all-healing). The Hippocratic oath opens with a rather pagan invocation to the whole bunch - Apollo, Asclepius, Hygiea and Panacea.

(That's not the only thing that's decidedly odd about the Hippocratic Oath. More on that at a later date...)