Horace is the Senior Wrangler of Unseen University. In the Roundworld, not only is a wrangler a trainer and handler of horses, but at Cambridge University, he is a graduate taking a "first" in mathematics.
The Senior Wrangler of Unseen University is very unlikely to be either a horse or animal trainer or a mathematician. Beyond a close acquaintance with the number 7a, who cares? What he nominally does is unknown. What he actually does is to eat several large meals a day and avoid students like the rest of the faculty. The title might refer to disputation of theses and judging of oral exams, or perhaps actual wrangling of creatures more dangerous than horses. Or perhaps he 'wrangles' the faculty members - keeping them in line for the Archchancellor. Since no one on the faculty does much to justify his title, his real duties are unclear. The Senior Wrangler is, however, one of the senior faculty and an academic who looks like a horse. A long-faced man who hates the sound of silence, his attempts to fill it up tend to do to conversation what it takes quite thick treacle to do to the workings of an intricate watch mechanism. In 'Hogfather', we learn that his first name is Horace.
The least corpulent of the senior wizards, mostly owing to his belief that long food (beans, celery, rhubarb) makes the eater grow taller. Several other foods (pineapple, clams, shrimps) he likewise avoids, because some of his less-fortunate relatives succumbed to fatal mishaps involving such comestibles. It's been mentioned that the Senior Wrangler has a poor sense of smell, and rather small feet.
Remarks he makes in 'Hogfather' imply that the Wrangler's family was poor, or at least frugal in their holiday celebrations. He may also be younger than his colleagues, judging by the Dean's snide remark about him losing his wisdom teeth "last week". Technically, the Dean outranks him at Unseen University.
He appears to be the most romantically-inclined of the major faculty given his deportment around Mrs Whitlow - especially since he once saw rather more than her ankles (see The Last Continent). He also has a rather ill-timed crush on the Cheerful Fairy in 'Hogfather.' During the Roundworld affair (The Science of Discworld), the Senior Wrangler championed the hypothesis that intelligence would most likely appear in the seas; earlier, he joined in on the student wizards' endeavors to hit "the Target" (Jupiter) with rocks, hence is likely responsible for one or more of that planet's moons. He was the last and most reluctant of the wizards to jump on the Dean's "Yo! Hut! Hut! Hut!" campaign in Reaper Man, but has since come to terms with himself a touch, and likes a bit of a righteous zap with his staff as much as the next man. Unless the next man is the Dean. No-one likes it as much as the Dean.