He took up the position after the previous bursar was killed in Sourcery. His name is Professor A.A. Dinwiddie, DM (7th), D.Thau., B.Occ., M.Coll., but, like most of the faculty, he is generally referred to by his title (It should perhaps be noted that he specifies that his name is "Dinwiddie with an O").
The Bursar is a quiet, reserved person, who took the job of University treasurer because he had an affinity for numbers (the Archchancellor describes him as "one of those idiot servants") and there was less competition for the role than other faculty posts. He expected to spend the rest of his life quietly adding up rows of figures. Unfortunately, shortly after he became the Bursar, Mustrum Ridcully was appointed Archchancellor. The brashness of Ridcully's personality wore away at the Bursar, a man whose idea of excitement was a soft-boiled egg, and Dr. Dinwiddie is now almost completely insane. He is kept functional, just, by experimental dosages of dried frog pills, though the effect is sometimes erratic. The pills are actually hallucinogens, the idea being that a proper dosage will cause him to hallucinate he is sane. An improper dose causes him to demonstrate symptoms of catatonia or disorganized schizophrenia, or cause him to believe he can fly. The last case is relatively easy to deal with; since the Bursar is a wizard, the other faculty members simply have to keep him from flying higher than the walls. In The Last Continent, we discover that he is an able surfer.
He has some talent for survival (see: Lords and Ladies and The Science of Discworld I & II) and is widely accounted as a neurovore i.e. someone living on his own nerves. For example, in Soul Music it was said he's been shot twice by arrows and survived. It has also been observed that no matter how detached from reality he gets, he retains his skill with numbers (in fact he is possibly the only person on Discworld who understands imaginary numbers, since much of what he's familiar with is imaginary). Hex temporarily inherited the Bursar's condition after a "conversation" with him, until Archchancellor Ridcully remedied the matter by convincing the ant-run thinking engine it had just been administered "LOTS OF DRYD FRORG P¼LLS".
The Bursar's insanity has become a byword in Ankh-Morpork; "to go Bursar" is "to go crazy." This is presumably a play on the term "to go berserk".
Questionably beneficial side effects of the Bursar's unique mental state are an ability to deal with certain types of bewildering or dangerous situations without batting an eye (though everyday events like Ridcully firing his crossbow across the hall and into a target above the Bursar's desk chair still put him on edge), and to occasionally perform impossible feats of ingenuity (for example, locking himself in the university's safe and somehow getting the key inside with him and locking himself in rooms that actually had no keyhole or key to start with).