An asklepian is a symbol of a rod entwined with a serpent, which represents healing and medicine.
You're probably thinking of caduceus, the staff with entwined serpents which was the symbol of the Greek god Hermes, the messenger, or asklepian the staff with one coiled serpent which was the symbol of the healer god Asklepius.
The doctors' staff is the Rod of Asclepius, Asclepius being a Greek god associated with medicine and healing. It has one snake and is not winged. The caduceus of Hermes has two snakes and is winged. In modern times, it is a symbol of commerce.