In Daggerfall there are seven different schools of magic; in skyrim there are five. The only explanation we are given is that they were moved to the other schools. But this doesn't make sense, how can different types of magic fit together? The style of the magic's have to be similar that's obvious, or do they? Could all magic be placed in one category? Why do different schools of magic even exist? My theory is without schools of magic learning spells would take longer; because you would have to master single spells at a time. With that in mind why not place all magic in one school? The answer is that this makes the spells harder to control. When atoms move they cause heat; the faster they move the hotter they get. To master a fire spell you would need to learn the exact speed the atoms need to move. Now to master destruction you would just need to learn how to make the atoms move faster and slower (and something else for electricity); not the exact speed the atoms should move. This would make learning a lot faster but the spells worse. All magic could be placed together but magic would be much harder to control. So what do you think?

The schools are an invention to make magic easier to learn. The powers that be decided that Mysticism was garbled, so they moved some of the spells to Alteration and some of the spells to Restoration.

It's like academic study. Science used to be considered philosophy and/or alchemy. Then they decided alchemy was stupid and philosophy would be more theoretical, while science proper would involve rigorous testing. Then it was divided into physics, biology, chemistry, etc...

Now, if you know chemistry you are also doing lots of math and should understand physics. You need a bit of chemistry to understand biology in any meaningful way. They do all go together, but they are separated to allow us to focus more deeply on one aspect or another.

The magic schools are different aspects of changing the world, but they are all fundamentally the same thing. Making your skin as hard as stone is the same as turning the air in front of you into fire. You are convincing the universe that what you want to be reality is actually reality. It's just easier to learn if you group similar spells into schools.

Magic effects are known properties of the manipulation of Magicka and spells are shapings of those properties. Response to Bero's Speech suggests that magical schools are just classification metrics, and that which effects fall into which schools is a matter of academic contention.

What everyone else said. The "schools" are just an invention by mortals that helps them to categorize and understand magic a bit better. Once upon a time there were no schools... there was only magic. I'm sure there are some academies or societies out there that have 20 different schools of magic (we just never experience it in-game). I like to think that the schools we know from the game came into existence with the Mages Guild. That would explain how it is seemingly universally accepted (due to the Imperial Empire's influence over the rest of Tamriel).