Krampus is an associate of Santa Claus. Like him, he is also in the Christmas business.
While Krampus wasn't allowed to perform the main gig (the last time he did, in 1911, this ended in a catastrophe the business took years to recover from), he occasionally handled extreme cases, like the one of Desmond Fishman, via appearing in one of Desmond's dreams. Desmond apparently was so far on the naughty side of Santa's list ("the worst boy"!), that "an emergency task force had to be assembled to judge [his] deeds". And Krampus was indeed successful in reforming Desmond - at least to a certain degree: The next morning, on Christmas Day, Desmond gave an obsolete coin to a "boy" for buying a turkey, and tried to shoot some crows in order to cook them as Christmas dinner for his butler Fotheringay.
Krampus was a furry beast with horns and a long tongue. His voice was once described by Desmond as "guttural".
Krampus' job was it to punish naughty children (and childish mer-men). According to Desmond's dream, one form of punishment (from a list of several the naughty child or mer-man was allowed to chose from) involved bodily trashing with birch twigs, and another one being eaten alive. (Amazingly, Desmond found the latter to be the more preferable option, though still a highly unpleasant one...)
He also seemed to be quite tech savvy, owning a tablet computer and having an own Pinterest account. He was only posting pictures of skulls, though.
In real life, Krampus is a character from Alpine folklore. You can learn more about him here.