It is rumored that Shalidor was the first to separate magic and spells into distinct schools, while some say that it was Vanus Galerion, founder of the Mages Guild. But, the first known written record of a proposal to establish distinct schools of magic in the Mages Guild was written by Gabrielle Benele of the Daggerfall guildhall sometime in mid 2E, who based it on a system that was used at the Shad Astula academy in Deshaan.
At their core, the schools of magic are not absolute and are simply artificial categories to aid in the classification and organization of magic and spells. In fact, some types of magic and spells may fit into multiple schools, and there are types of magic that aren’t encompassed by them, such as alchemy and enchanting, that are magic nonetheless.
Over the eras, the schools themselves have changed, with two having been removed and consolidated into other schools by 4E. The school of thaumaturgy was dissolved before the end of 3E, and the school of mysticism some time in 4E.
Further Reading:
Proposal: Schools of Magic, Gabrielle Benele
This graphic uses The 6 Schools of Magic Resources by jeclxohko on the Skyrim Nexus