It's like this.
The practice of exorcism is known in one variant or another, to all religions both on the Disc and on Roundworld. It is closely related to Banishing rituals: the idea being that the practitioner of magic or religion must be able to put down again that which he calls up, otherwise that hapless practitioner is in what Albert calls "the gyppo", mystically and metaphysically speaking. (Or, more specifically to Christianity, the forcible removal of unwanted spirits through the power of God.)
The idea that the human mind and soul may be co-inhabited, even forcibly occupied, by another is a widespread and prevalent one. There is even lots of anecdotal and actual proof for this on the Discworld: regard the practice of Borrowing which is a key and much admired proficiency in Witchcraft. Or at least, a much envied and much-coveted proficiency, by those witches who don't have the knack. This is a benign and largely benevolent form of possession - in fact, it cannot even be called possession, as a skilled witch such as Granny Weatherwax makes pretty damn sure that she is riding in another creature's mind purely as a passenger, albeit one who might from time to time make a suggestion to the driver as to speed and direction, as befits a back-seat rider. For a Witch knows enough to be absolutely sure that she has to be principled about how she steers the mind of another - we have a damn good reason not to go to the bad, as another practitioner of Magick said*. Granny is also aware that there must be some sort of pay-off to the creature which is Borrowed, offered by way of compensation: if she Borrows an eagle so as to be able to patrol Lancre from the air and reassure herself that all is well in the kingdom, she will make pretty damn sure there is a substantial meal of rabbit chunks awaiting it afterwards. A true possessing spirit wouldn't bother saying "thank you", nor would it go to the trouble of making you a dinner by way of compensation. As Theda Withel was told in Moving Pictures when her body was temporarily hijacked for other ends, a real evil spirit wouldn't have let her put her shoes on first, and wouldn't have cared what it made her walk across.
Trouble arises when a self-taught witch walks in and out of her body at will, and considers this to be such an unremarkable thing that it isn't worth mentioning to her teachers in the Craft. Who, had they known, might have informed her that this is something no witch ever does frivolously, and whenever she does it, it is vitally important to leave your body guarded and protected while you are Out, otherwise Something Else might find the metaphysical loose windowcatch or back door key under the doormat before you can get back. In other worlds, this sort of thing inevitably leads to three hundred and sixty degree rotation of the head, accompanied by copious quantities of pea-green soup. Well, that's optional for demons, although Crowley would have too much style for that - he'd leave the gross-out stuff to Hastur or Ligur - and totally unspeakable for angels.
And then there is the sad but curious case of Altogether Andrews, described as some innocent but hospitable person of a psychic disposition who was simply overwhelmed by eight colonising souls, who take it in turns to possess his mind and body...
We also have Miss Pointer and Miss Pickles who inhabit the same body, but only Miss Pointer is aware that they rent out their basement to Mr. Shine's Thud academy...
It is possible to be possessed by an Angel of the Light, although most religions use a different vocabulary for this sort of thing, choosing to describe it as Walking With the Holy Spirit, Being Exalted, Yea, Even Unto The Seventh Heaven, and "loas, be buggered". The three or four people serially possessed by a temporarily discarnate Aziraphale might have a different word for it, calling it, for instance, a bloody nuisance, or "why am I talking like a poofter, if I'm communing with the spirits of the ancestors?"
Just as the Omnian Quisition had its executive arm, the Quisition, and just as before you can exhume a corpse, the Assassins' Guild might first have inhumed its former owner, possession, and the subsequent option of exorcism, also has its flipside.
Insorcism is not the practice of taking an unwanted (not necessarily evil) spirit out of a person or place it is lodged in, often quite happily. In the hands of Professor John Hicks and the Department of Post-Mortem Communications - not, we wish to make it clear, a University department which has ever in the course of its life had anything at all to do with necromancy - insorcism is the precise opposite. Before exorcism can happen, somebody has to put the spirit there in the first place. The job of the Insorcist involves placing a discarnate spirit in a lodging from which it subsequently will need to be winkled out with the metaphorical equivalent of a very big pin - it helps if there has been some negotiation between the Insorcist and the spirit as to selection of a preferred dwelling-place beforehand.
Thus did Professor Flead find himself living a happy afterlife in the Pink PussyCat Club, well out of the Department and way out of Hicks' hair... much to the consternation of the club management who cannot get anyone to sit in Seat Seven (front row, centre) for longer than nine seconds. They did try to take the seat away - but not for very long....
This is such a potentially potent skill that the University has banned the practice on ethical grounds. Otherwise, the general Ankh-Morpork ethos might come to the fore, even among Wizards, and a professional Insorcist could find himself with a lucrative living... "I hear you've got an indwelling psychic entity, squire. I can get it out, but you need to look at my scale of fees first..." Or "Nasty things, these possessing spirits. Truly nasty, friend. How many of 'em wouldn't you like?"
Therefore to buy the silence of his undergraduate student wizards, Hicks has to promise doctorates to a typical group of student wizards who, in any given term, only manage to present two-thirds of an essay between them. After all, as Hicks says, the College Council was perfectly correct to ban Insorcism on the grounds that it might fall into the wrong hands - ie, anyone else's but ours!
The whole business of Borrowing, possession of varying degrees, and Insorcism in particular, are dealt with in various books in the series; Insorcism as a specific concept is a feature of Making Money.