The symbol is called a Caduceus (heralds staff).
"In Greek and Roman mythology, a magic wand consisting of a rod topped by wings and intertwined by two snakes (kerykeion, caduceus in Greek; kerykeion skeptron meaning "a heralds wand"; keryx meaning announce or herald) was depicted as a medicinal or magical tool symbolized to indicate healing and immortality in literature and drawings from the era before Christ. The fabled wand or rod, the caduceus, was carried by Hermes in Greek myths and Mercury in Roman mythology as the messenger of the gods. Originally, the caduceus was represented as a simple staff wound about with two white ribbons. It is a figure that consists of two entwined serpents encircling a wand or rod. It was a symbol of authority and inviolability and protected the herald who carried it. In Homer's Lliad and Odyssey the caduceus is often mentioned as a type of magic wand by which Hermes opened and closed the eyes of mortals. It was therefore connected with death and the journey through the underworld. Later myth says Hermes once threw his magic wand at two snakes fighting on the ground. The snakes became entangled in the the magic wand and have been attached to it ever since. The wings at the top were added in later Greek and Roman art. In Virgil's Aeneid (book 4) the caduceus is said to have been given to Mercury by Apollo in exchange for Lyre. Milton, calling it Hermes' "opiate rod" in Paradise Lost (book 11.133), refers to the belief that the caduceus is associated with medicine because it was one of the symbols of Aesculapius, the god of medicine for the ancients. Le Sage, in Gil Blas (1715) writes; "I did not think the post Mercury-in-chief quite so honorable as it was called... and resolved to abandon the Caduceus [give up the medical profession] for ever."
Some confusion exists between the wand carried by Hermes and the Staff of the demigod Aesculapius which is classically characterized as a single serpent encircling a rough hewn tree branch. The Staff of Aesculapius is truly the more legitimate symbol of medicine, however the Caduceus has been adopted as the more commonly used symbol in medicine.
According to mythology, Hermes threw his magic wand at two fighting snakes. The snakes became entwined as they stopped fighting. The actual origin of the Caduceus is from two sources. The first was from the Babylonia god Ningizzida and the second was from a shepherd's crook that was forked on top."