While the men parade around dressed as creepy demons, the women get to have some fun too, wearing masks and representing Frau Perchta, who was a Nordic goddess, who may have been an aspect of Freyja, the fertility goddess. Interestingly, in the Pennsylvania Dutch community, there's a character called Pelsnickel or Belznickel who is an awful lot like Krampus, so it appears that the tradition migrated across the water when Germans settled in America.
Krampus.com, which calls itself the official home of "Krampus, the holiday devil," calls Krampus a "dark counterpart of Saint Nicholas, the traditional European gift-bringer who visits on his holy day of December 6th... The bishop-garbed St. Nicholas rewards good kids with gifts and treats; unlike the archetypal Santa, however, St. Nicholas never punishes naughty children, parceling out this task to a ghastly helper from below."
-The Crafty Witch