The Krampus is a companion of Father Christmas in Germanic Christmas celebrations. The Krampus is also demon that whacks the shit out of kids for being naughty.
It's Christmas morning in Germany. You wake up in your little bed, shake the sleep from your eyes and creep downstairs so as not to wake your Mutter und Vater. You see that all of those schnitzels you left for St. Nick are gone and there's a big pile of presents under the tree and the stockings all along the fireplace mantle are filled with candy and toys with small parts you fully intend to try to swallow later. But as you shuffle in your footy jam-jams to the presents, you feel a series of blows to the back of your head and knees. You crumple to the ground, sobbing and soiling yourself from fear. Then a hideous horned creature glares at you through matted locks and snarls. You know that God has sent this beast to punish you for your sins and you prepare to be thrown into his basket to be dragged off to the Pit to burn for dipping little Leonie's pigtails in ink. And the masturbation. Just so much time wasted slappin' your strudel. Then the creature removes its face and you see that it's just your Uncle Friedrich and then the whole family comes out laughing like it was all a joke and then they give you sweetie, cause candy makes getting beaten by a monster all better. Congratulations, you just celebrated Christmas in Germany.
The Krampus is a demon that follows Santa as he goes from town to town. In most other parts of the world, Claus usually just gives you a lump of coal for being a little dick, but if you live in Italy, Germany, Austria, the Czech Republic, or Hungary you get beaten by a flippin' DEMON FROM HELL who then stuffs you in his basket to be tossed into the Malebolge.
Typical reaction to hearing of the Krampus
The tradition in countries where the Krampus is popular is for young men to dress as the monster during the first two weeks of December and to scare women and children by chasing them with their birch rods and rusty chains. There are actually competitions to see who can build the most horrifying costumes. All we have in the States is to see who can be the asshole with enough lights to seen from space.