This is known as the Rod of Asclepius. It's an international symbol of healing.

You will find it on ambulances inside a blue, six-pointed geometric figure known as the Star of Life. A modified version of Hermes' messenger staff, with two snakes intertwined around it and wings on top (called the caduceus) is the symbol of the American Medical Association.
I think symbolism is cool. It's like, I dunno, taking a thousand words or so and boiling it down to one picture.
But why is a snake on a stick the symbol of healing? (Sounds like a Louisiana bayou delicacy, to me)
Asclepius was probably a primitive doctor (redundant, I know!), who the Greeks started worshipping as a god after their conquest of the world, circa 300 B.C.
Over 1,000 years
before Asclepius, however, there was a snake on a stick that actually
was the agent of healing.
Near the end of their wandering in the desert, the Israelites were complaining (again! - y'know, I personally prefer people who would honestly reject my gifts; rather than accept them, only to whine and moan about them), so
fiery serpents were unleashed on them and Moses had to plead for them yet again. (Note: I found the linked article in
Meridian Magazine, a Mormon publication of all things! Interesting information, though)
God told Moses to make a snake and put it up on a pole and whoever looked on it after they were bitten, would live (Numbers 21:6-9).
Way before that, while Moses was on the lam from Egypt after killing someone and just cooling his heels in the desert, the Lord spoke to him from a burning bush (Exodus 3). He was actually watching his father-in-law's sheep at the time, probably because his in-laws were busy metalworking. His father-in-law, whether named Reuel, Jethro, or Hobab (the Hebrew word translated
father-in-law just means a relative by marriage) was not only a priest (this obviously was before Levitical law,
a la Melchizedek) but a Kenite (Judges 1:16). Now, Kenites were a nomadic tribe of coppersmiths (there's quite a bit of copper and manganese in the Mt. Horeb area - they're the principal ingredients of brass, y'know). Moses begged them to accompany the Israelites, because of their desert-living experience (Numbers 10:29-31).
So (thanks for bearing with me, so far), Moses' relatives probably whipped together a brass snake with facility.
Everyone knows the account of the serpent in the Garden of Eden - it says the serpent was more crafty (or subtle or cunning or shrewd) than any other animal (Genesis 3:1), and if we couldn't figure out who that serpent was, the last book of the Bible (Revelation 12:9) dispels all doubt. Interestingly, the instructions Jesus gave when he sent out the Twelve were "I am sending you out like sheep among wolves. Therefore be as shrewd as snakes and as innocent as doves."
Jesus referred to the event of
the snake on a stick (thus imparting its true symbolism) when he talked to Nicodemus at night in the most quoted passage in the Bible, John 3:14-16:
"And as Moses lifted up the serpent in the wilderness, even so must the Son of man be lifted up: That whosoever believeth in him should not perish, but have eternal life. For God so loved the world, that he gave his only begotten Son, that whosoever believeth in him should not perish, but have everlasting life."
So, the symbol of sin became the agent of healing.
He made Him who knew no sin to be sin on our behalf, so that we might become the righteousness of God in Him. 2 Corinthians 5:21.
The ultimate fate of the brass serpent came 686 years after the original event, when Hezekiah became king of Judah.
He removed the high places, smashed the sacred stones and cut down the Asherah poles. He broke into pieces the bronze snake Moses had made, for up to that time the Israelites had been burning incense to it. (It was called Nehushtan.) - 2 Kings 18:4.
Apparently, the Israelites were worshipping the thing.
They exchanged the truth of God for a lie, and worshiped and served created things rather than the Creator - Romans 1:25.
Anyway, it comes as no surprise how the Greeks came up with their symbol for healing. Especially because it was only a couple hundred years since the brass serpent had been worshipped (and it must have had quite a bit of word-of-mouth currency). I mean, after 2,300 years, we're still calling it the Rod of Asclepius.