In Celtic mythology, snakes were symbolic of many things. Snakes were a symbol of knowledge, cunning and change. This idea of transformation most likely is the result of the snake's ability to shed its skin leaving part of its old self behind. So, snakes are also considered to be symbolic of rebirth in the Celtic culture.
In the keen Celtic mind, snake symbolic meaning of transformation came from the shedding of its skin. Physical evidence of leaving its form behind (casting off the old self), and emerging a sleeker, newer version made the snake a powerful symbol of rebirth and renewal.
The serpent and snake in mythology has had many meanings. It has stood for both good and bad, life and death, and in many instances have stood for fertility. The shedding of their skin has also been associated with immortality.
At least in Western mythology, serpents are usually seen as bad things. The most obvious example of this comes in the creation story from Genesis in the Jewish and Christian scripture. In that myth, the serpent is portrayed as a very sneaky and cunning animal who tricks Eve and leads to the "fall" of humanity.