It's ouroboros. It's used to symbolize many things, including unity, infinity, and the cyclical nature of life. One of my best friends uses it to represent his psyche.

Are you referring to two snakes swallowing each other or the snake that's swallowing its own tail? The former has no common name with which I am familiar. The latter is called the "ouroboros" and has roots in many different ancient civilizations, some of which were never known to interact. The ouroboros is found in Aztec, Middle Eastern and Native American cultures. The European form has its origins in Ancient Egypt. The symbol traveled from there to Phoenicia (modern Israel) and from there to Greece. The Greeks, although they did not create the symbol, gave it its name. The ouroboros represents a cyclical nature, a dual nature or complementary opposites, depending on context and culture. The symbol was popular in alchemical contexts and a fixture in Greek (and likely other cultures') magic. http://en.wikipedia.org/wiki/Ouroboros

A caduceus (/kəˈduːsiÉ™s/, -ʃəs, -ˈdjuː-; kerykeion in Greek) is a (sometimes) winged staff with two snakes wrapped around it. It was an ancient astrological symbol of commerce and is associated with the Greek god Hermes, the messenger for the gods, conductor of the dead and protector of merchants and thieves. Among the Greeks it was originally a herald's staff, sometimes with wings, with two white fillets of wool attached. The view of Karl Otfried Müller, that the ribbons eventually evolved into snakes, was held for several generations of conservative mythographers, though Jane Ellen Harrison detected that Hermes had originated in snake-form and that the snakes were essential to the caduceus, though she was unaware of the Near Eastern connections. The caduceus is sometimes inaccurately used as a symbol for medicine, especially in North America, but the traditional medical symbol is the rod of Asclepius with only a single snake and no wings. http://en.wikipedia.org/wiki/Caduceus In ancient Greece the sign for the planet Mercury was drawn 5017. In Greece Mercury was represented by Hermes, the quick-footed messenger of the gods, often pictured with wings at his ankles. Hermes was symbolized by the caduceus, the staff of the two snakes 41b16b. The god in the Roman empire that the messenger of the Greek gods, Hermes, was identified with, Mercury, was the god of merchants, but also of thieves. The staff of the snakes in Rome became a symbol for trade and communications. This caduceus, or snakes' staff, has been mixed up with the Aesculapii staff, with only one snake, a symbol for the art of healing or medicine. Aesculapius for the Romans and Askleipos for the Greek, this god was the son of Koronis and Apollo and the god of medicine and healing, worshipped at Epidaurus. Zeus killed him with a thunderbolt for restoring Hippolyte to life at the request of Artemis. http://www.symbols.com/encyclopedia/41a/41a12.html The symbol's origins are thought to date to as early as 2600 BC in Mesopotamia. http://www.crystalinks.com/caduceus.html The symbol of two intertwined snakes appeared early in Babylonia and is related to other serpent symbols of fertility, wisdom, and healing. http://www.vibrani.com/ccexperiment.htm In Kundalini Yoga of Hinduism, the symbol of two snakes starting from a circle is related to awakening of wisdom and healing. http://www.search.com/search?q=snake+symbolism+in+Kundalini+Yoga Indians were the originators of many aspects of scientific knowledge. The knowlege of Kundalini originated in ancient India and spread to the west. http://en.wikipedia.org/wiki/Science_and_technology_in_ancient_India