What's an Ouroboros, anyway?

uroboros. Also ouroboros, uroborus. The symbol, usually in the form of a circle, of a snake (or dragon) eating its tail.

--The Oxford English Dictionary, Second Edition

The Ouroboros is a symbol of renewal, infinity, and the Eternal Return (as in Neitzsche's philosophy and Poincaré's recurrence theorem for systems with finite and bounded phase space).

The snake biting its own tail image used in the opening title sequence of Millennium is called an Ouroboros. In the series it is the symbol of the Millennium Group. In mythology, the Ouroboros is any image of a snake, worm, serpent, or dragon biting its own tail. It was first seen as early as 1600 years BC in Egypt.  The Greeks called it the Ouroboros, which means "Tail Eater."  Generally taking on a circular form, the symbol is representative of many broad concepts.  Time, life continuity, completion, the repetition of history, the self-sufficiency of nature and the rebirth of the earth can all be seen within the circular boundaries of the Ouroboros.  Societies from throughout history have shaped the Ouroboros to fit their own beliefs and purposes.  The image has been seen in Japan, India, utilized in Greek alchemic texts, European woodcuts, Native American Indian tribes and even by the Aztecs.  It has, at times, been directly associated to such varying symbols as the Roman god Janus, the Chinese Ying Yang, and the Biblical serpent of the garden of Eden. 

Chris Carter once explained in his own words, "The image that appears on the main title of the show which is a snake eating its tail, a circular snake, the snake is a classic and ancient image. I think it's come from several different cultures, it's called the Ouroboros and it has many different meanings. One is eternal return, everything being circular, everything recurring and also of a snake eating its tail, of it devouring itself, which is a more negative context. I thought for Millennium it had equal parts of each and left some then to the imagination of what it might mean. But it's a very powerful image and I think it's really perfectly representative of the show."
 

Find out more about the Ouroboros here.