Magic in fantasy is like physics in science fiction. There are rules, and then you set about cleverly breaking or bending them, without destroying the whole system. In science fiction, this is usually done by describing some method by which faster than light space travel is possible. Any space-faring version of humanity either needs to figure out how to travel faster than light, or needs to have a method of space travel that accounts for the vast amount of time it would take to travel to even our closest neighbors in the cosmos.
In worlds of fantasy, there is often a system of magic, and within this system there are certain rules about how magic works. Unlike physics, which has rules that are agreed upon in the real world, magic has rules that are determined by the author of the story. Authors have the choice to either be very vague about how it all works, or very specific. However, it's important for the integrity of the story that the system function according to whatever conventions the author sets forth, and if those conventions are broken, it's purposeful and not accidental.
Let's take something like the Dragonlance Chronicles. Not the best written series by any means, but fun, high fantasy adventure. Reading them is like reading the description of a Dungeons and Dragons session. The rules about magic in Dragonlance are simple, but powerful. Basically, magic is hard to hold in your mind. So, magicians spend part of each day memorizing the spells in their spellbooks. This is exhausting. Once they've used a spell, it's gone from their mind and they have to spend time memorizing it again. This sets up various possibilities for tension in the story: What happens when a wizard has said all their spells? Where can a wizard learn new spells? Is there ever a time when a wizard remembers a spell without having to memorize it again?
Another interesting system of magic is from The Bartimaeus Trilogy by Jonathan Stroud. In that series, 'magic' has nothing to do with spells, or inherent power or anything like that. Magicians learn to summon spirits and geni of different levels of power. If they do it correctly, the spirit must do their bidding. Magicians can see these spirits, or at least, they can see some of them, while ordinary people can't. So, when a magician puts a spirit to work, ordinary people see the outcome as magic. If the magician does not say every word precisely right, or make every drawing just so, demons can escape from the confines of the summoning, and kill or enslave the summoner. This creates some great tension for the story, and plenty of options for interesting storytelling. Every summons is a risk, and everything said by a summoned demon is questionable, because they're always trying to trick their summoner into making a mistake.
In Harry Potter, there is magic. That is evident from the get-go. How magic works is not entirely clear. All the things that belong in the magical world have been separated from all the things that don't belong in the magical world. Who decided what was magical and what wasn't is not entirely clear. Wizards and witches can use magic. Magic is something you are born with. Growing up, weird stuff will happen around you. If you're in a muggle household, this weird stuff might make you or the people around you think you're going crazy, but no one from the magical world is going to tell you anything until you turn 11 and get invited to go to magic school. On the other hand, if you grow up in a magical family, then you know all this from the start. That kind of sets up an unfair advantage at school for those born in a magical family. This dichotomy is never really explored in the books, except through some of the statements by Draco and friends about purebloods being better than mudbloods. And, if you're magic and you have magic and your magic could destroy stuff on accident, and if magic happens in the muggle world then you get in trouble and people's memories need to be wiped, then it would make a lot more sense if magic users were identified when they are very young, and the kids and their families are brought along in some way so that when a child accidentally blows up their aunt, or their hair won't stay cut, the parents/caregivers won't start trying to get an exorcism or put the kid in an asylum. Which is what would happen in this world, at least some of the time. Other parents would try to keep it hidden, or make their children feel ashamed of it, or, conversely, would seek to capitalize on it, taking their kid on the road and making a side-show of their abilities. But that doesn't happen for some reason in Harry Potter.
If you're going to do magic on purpose instead of by accident, you need a wand. Wands are all unique-ish, and 'choose the wizard.' This suggests that every wand in a wandmakers store has an eventual wizard partner. With only 40 new wizards per year, and about 3,000 wizards in all of England and Wales, it seems like it would be hard to make a living as a wand-shop owner. You've spent all your capital on a massive quantity of stock, but you only have customers about one week out of the year: when 11-year-olds are buying their first wand. The rest of the year, you might have a handful of customers - wizards who have lost or broken their wand, and need an new one to choose them. It seems like, from a business perspective, a wand-shop would need to have other goods or services to carry through the rest of the year. In addition, Hagrid suggests through his recommendation of Olivanders for wands that there may be other wand sellers someplace. If that's the case, then the right wand for the wizard could be at a different shop. What if the right wand for you is at the shop in Edinburgh? Or Paris? Alright then, I guess, like love, there could be an array of wands that are a good fit, and some might end up being really good fits while others are marginal. In the case of Ron Weasley, just use a hand-me-down wand. How the hell does that work? The wand was originally owned by his older brother Charlie. Does that mean that it chose Charlie, but only for a while, and now the romance is gone? Or did Charlie ditch it when it got old, and decided to get something newer and nicer. Do cheap wands pick poor wizards? Then there's the super wand that we learn about in the later books, that makes you invincible if you're its owner. And then there's the whole complicated logical acrobatic argument that Rowling takes us through to prove who owns the wand, based on new rules she thought of as she went along that sort of work with the assertions she made in the first book.
In addition to the wand, you need to learn spells. You learn the spells, but that's not enough. You also have to really mean it when you say a spell. That's fine. Once you've learned a spell, though, you've got it and you can use it whenever. There doesn't appear to be a personal cost in energy or sanity or anything to use a spell, once you've learned it. This suggests that magic has an infinite source, or at least, there is more magic than wizards could ever use in their lifetime.
Wizard + wand + spell = magic happens. However, this isn't always true. Without the wand, you can still do magic, it's just not very focused. You can put the broken pieces of a wand in an umbrella and secretly do magic, even if you're not supposed to. You can use enchanted objects, like brooms, without using your wand. You can interact with magical beings without your wand. You can do potions without any silly waving of wands, except when you have to wave your wand to finish the potion. You can do magic with mathematics. We call it calculus. You can do magic without a wand by seeing the future. A wand seems more like a jedi lightsaber. The preferred tool for the job, but not the only one.
If I had to guess, I would say that Rowling didn't want there to be any limits on what magic could do, so she tossed every kind of magic she'd ever heard of into the pot, and called it Hogwarts. This is still fine - I like that it's all possible. However, it also means that anything you didn't set up as 'normal' magic in the first book needs to be justified in later books to prove that it's normal, too. Some people don't have to speak the spell out loud to magic the magic work? Oh, well that's something you learn to do when you're older. Also, things that appear in the books that are actually extraordinary, but get presented as normal at first, need some kind of justification later. The invisibility cloak is actually a unique item. Ron says "they're really rare," when in actuality there's only one, and Harry has it. If they weren't really rare, then that would mean other people could have them and use them, and chaos, man! Chaos! So let's back off on the availability of invisibility cloaks. Why is a spell to breathe under water so hard to come by, while other, less useful spells are really easy? Look, mom, I learned to summon a large snake! Lovely.
It would be really easy to set forward a basic system of magic, with simple to understand but powerful rules, right off the bat, by having Harry learn about them in his first class. It would then be fun to find the ways to bend or break the rules - the Weasley twins would dedicate their lives to hacking magic, I'm sure. The Slytherins would have a high interest in sneaky ways to use magic to get what they want without actually breaking any laws or rules. A system of magic gives characters something to act within or against. If it's thoughtful, it can carry a whole series of books without limiting the storyline, and instead of being a clunky system with junk added to it every book so that fun stuff can happen, it can be a device that helps drive the plot and create great tensions.
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