A friend of mine suggested to me in a conversation this weekend that part of his trouble with the Harry Potter books was that there was no "theory of magic" in Harry Potter. No one, he argued, could construct a general theory of how the magic worked; it appeared to him to be a purely ad hoc mechanism.
I think he is mistaken, and there are definite rules and fact appertaining to magic in Harry Potter, and what is more, those are the keys to understanding some of the major themes in the books.
I will admit that it is true that not all the magical methods are introduced at once. There is a degree in which it is true that elements of magic are introduced as required. For example, the Portkey, a charmed device used to transmit wizards from one location to another, first makes an appearance in The Goblet of Fire. But there we are told how it works, introduced to it at the start of that book, and like the proverbial shotgun mounted on the wall, it makes a crucial re-appearance at the end of the book. But the core of Voldemort's quest for immortality is there throughout the books, even before the Horcrux is given a name.
The key to understanding magic in Harry Potter is that it is genetic. The magical ability is transmitted through wizarding families. Parseltongue, the ability to speak the language of snakes, is also passed down the family line by inheritance. But there are exceptions - there are rare mutations, where the magical ability emerges from non-wizard families, and it is clear that wizarding may have come about from these kinds of mutations, with selective breeding amongst those with magical abilities, thereby breeding for the magical genetic inheritance. On the opposite kind of this are "squibs", those born to magical parents but who do not possess magical ability. They are part of the magical world, because their parents are, but while they can understand how spells are cast, they cannot perform magic, because that requires the inherited ability.
So this is a very Darwinian framework, and within that framework, racism rears its ugly head. There are wizards who pride themselves on their pure blooded ancestry, and who look down on the mutants (those born to "Muggle", non-magical parents) as "mudbloods". The theme of racial purity drives the narrative, with Lord Voldemort pushing for the extermination of the impure "mudbloods", as well as the eventual enslavement or extermination of the non-magical peoples. The Nazi extermination of the Jews for racial reasons, ethnic cleansing, and all these aspects of racism come into the struggle between the two sides in the Harry Potter books, because of the Darwinian framework.
But there is one more aspect to magic. Magical ability for wizards depends upon wands; without a wand, magic can be performed, but none of the really powerful magic. That depends upon a wand, which is used to cast a spell by an incantation. That can be spoken out loud, or it can be internal speech, where the words are thought, not spoken; the latter is more difficult. All use of spells requires practice, it is a skill, which has to be honed to be automatic. It is possible to create new spells, but the suggestion in the books is that this is rare - spells come and go as fashions. Nevertheless, like experimental scientists, new spells can be created.
Wandlore has it that "the wand chooses the wizard", and not actually vice versa. The wand then acts in a symbiotic relationship with the wizard, empowering the magic if it is a wand that has chosen the wizard, or transferring allegiance if the wizard overpowers another. But a wand that has one owner that has not transferred allegiance in this way will not perform as well as it should; the wizard knows that there is something deficient in their ability to cast spells. Drawing on biology, we can say that wands are non-sentient symbiots, which a wizard needs to be a full wizard.
Magic does obey certain rules. It is possible, for example, to prevent apparating or disapparating from a location - as with Hogwarts, or the cellar prison at Malfoy Manor. But that is wizarding magic, and it is established that house-elf magic works on different premises. House elfs do not need wands to perform powerful magic, and they can perform apparating or disapparating from locations where spells prevent wizards from doing so. House elfs are magical creatures, there are no non-magical house elves, and no non-magical mutations, and they are a different species from human beings. But they can be killed by non-magical means, such as a dagger, so are not invincible. Magic is different with different biological species.
The use of magic creates a kind of disturbance which can be tracked, regardless of whether it is wizard magic or elf magic. This is used primarily to detect use of underage magic, but can also detect unauthorised magic against non-magical people (ordinary humans), or if refined, even the use of Lord Voldemort's name.
One thing magical cannot do, and that is to bring back the dead. This is established very strongly throughout the series, and the fear of death is the driving force behind Lord Voldemort's bid for immortality, to safeguard himself from death. The acceptance of death is part of the reason why Dumbledore's scheme triumphs in the end, when Harry sacrifices his own life.
Sacrificial love also has a great magical power, and this ability to lay down your life to protect another's life is something which enchants the blood of the one so protected. Here we see a very Christian theme in Harry Potter, where sacrifice is linked magically to blood. Just as the Horcrux is an object which holds a portion of the soul of someone, and keeps them from death while it exists, Harry's blood protects Harry's life from Voldemort's attempts to kill him. One kind of magic - the Horcrux - is created by murder - the other kind, the enchanted blood - is created by sacrificial love. There is a mirroring here of the kinds of magic at its extremes - hate and love. These are the most powerful kinds of magic, and their magic is based upon motivation. But this also exists to a lesser kind. Harry is told that for the forbidden curses to be effective, "you must really mean them". So magic is not just incantation and innate ability, there is also an effort of will involved.
In conclusion, magic in Harry Potter does have a logic to it; for wizards, it is rooted in a Darwinian framework of selective breeding and mutation, and it is that which determines whether an individual can perform magic or not, and also (as one might expect in an evolutionary framework) why some wizards are more powerful than others; difference exists with magic, just as it does with sporting prowess, because it is part skill but also partly the cards that biology deals the individual.
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