Hogfather
The Hogfather is originally a winter god. In the deep winters, people of old feared that spring might never again come, so they gave bloody sacrifice of hogs to the Hogfather. Nowadays, the Hogfather is expected to travel in a sleigh pulled by hogs and every Hogswatchnight to bring gifts to all the children, or at least, those who believe in him; this yields a worshipper range of most of the Discworld except for the Counterweight continent and the continent of Klatch. Despite this impressive range, crass commercialization of the holiday compromised the belief in Hogfather, and then interference from the Auditors of Reality severely threatened his existence (chronicled in Hogfather).
His residence is the Castle of Bones near the hub. The Hogfather is an ancient being, who has seemingly kept his present form for some time; in Hogfather, Albert reminisces about his childhood memories of the Hogfather, more than 2000 years ago. The only difference in the Hogfather in that time seems to have been that he did not bring presents, but sausages and black puddings if you were lucky. But you always got a pink sugar piglet in the toe [of your stocking].
He parodies the Roundworld Father Christmas (the UK version of Santa Claus)
In the land of pine trees and deep snow and long winters, when the sun is below the horizon most of the day, in the piercing cold it can become a real question whether or not the sun will rise again in the morning. The voice of reason says it will rise, but there are many unreasonable voices in a person. In response to this question, the anthropomorphic wossname called the Hogfather arose. He had many forms throughout the centuries, and the man in the red suit was only the last of them. Susan saw them one after another. His first form was properly speaking porcomorphic, or pig-shaped. It was to this point that he retreated when ambushed by Mr Teatime. Despite his apparent task of distributing presents in the season of goodwill, sustained by the belief of children, on the longest night of the year it is still the role of the Hogfather to ensure that the Sun rises in the morning (comparable to Teppic’s dynasty). It is no small thing to make the Sun rise. When he was able to resume his role, the Hogfather seemed to salute Death as an equal. He did not thank him. “Ho Ho Ho,” notwithstanding, he was not a personification of many words.
When the Hogfather was attacked, Death, like an expert mechanic hearing a change in the sound of an engine, had heard a harmonic change in the music of the universe. He was able to enter the congruent reality of the Hogfather in a way in which he could not enter the domain of the Tooth Fairy. Death applied first aid to sustain belief in the Hogfather among children, and recruited Susan, even Hex.
When Mr Teatime had failed, the Auditors attacked the Hogfather directly, in the form of hounds. It was necessary for Susan, as a human being, to be the one to act to save him. To do this she had to leave behind her inner babysitter, and be with her belief. It was not simple. At a point where the hog she was riding teetered on an icy ridge, precipitous drops on both sides, she repeated to herself, “He’ll catch me if I fall,” but an older voice within her said, “No, he won’t. If I fall now, I don’t deserve to be caught.” As in the case of Death and Miss Flitworth, an anthropomorphic personification was saved by the self-gift of a human being.
Later Susan said to Death, “The sun would have risen just the same, yes?” “The sun would not have risen,” he said. “A mere ball of flaming gas would have illuminated the world.” It would have been a world of oblivion.
The same human capacity to believe which supports the Tooth Fairy and the Hogfather creates the space for meanings such as justice, mercy and duty, which have no existence in the physical universe, to become. What Hogswatch seems to stand for is not that you will get presents if you have been nice, but that what you have done counts for something, and it is known. It matters. “All things strive...” says Hex. Strange though it may seem, given the general immersion of human and human-like beings in the mere business of staying alive, it is towards this point that all things strive. This very line also appears in Dwarf Lore.
Annotation
Some aspects of the Hogfather suggest the anti-Santa Claus who lives on in some German folklore. Yes, Santa delivers presents to the good little boys and girls; but the bad ones had better beware the entity known as Schmutzli or Krampus, who at this season has free rein to punish the naughty with pinches, prods, hair-pulling, or a damn good thrashing with chains. Or even a "gift" of bloody bones in a bag... (thanks to Stephen Fry and QI for information)