The morphogenic field is the field that defines an organism's shape. This means that in order to change a creature's shape a magic user has to change how the creature thinks of itself. While this is possible for animals and other thinking creatures, it is much harder for plants and other non-thinking organisms.
The strength of a person's morphogenic field governs how long that person will potentially "survive" as a ghost after death.
Once the morphogenic field is tweaked for the first time, it will be much easier for the subject to be reverted (or revert spontaneously) to the modified shape again.
The most famous example of a redefined morphogenic field is, of course, the Librarian at Unseen University, who has been an orang-utan for so long that he now defies all attempts to turn him back onto a human.
See also Morphic resonance.
In biology, the word "morphogen" means, very roughly, "a chemical which induces growth into a particular shape". The precise definition is unrelated to "morphogenic field" as used in the Discworld series and therefore beyond the scope of this article.