While the HP universe doesn't go very deep into the "metaphysics of magic", there is a layer of consistency in the way that magic works. Specifically, magic involves what you think and what you do, across the board. Wand movements act as physical focus points for the 'magical energy', while the incantation acts as a mental focus point. We see spells in both English & Latin, but never hear what the Beaubaxons, the Durmstrangs, or those from the American Salem school, use for their spells. They may be the same, they may be different, and either way, the incantations still carry the same purpose - focusing the mind of someone who has the magical ability. Since WE don't get to learn how it works, we have to observe how people in the universe interact with magic, and how magic interacts with them.
- When HP enters the Floo Network, thinks about going to Diagon Alley, but says "diagonally", He shows up in a fireplace in Knockturn Alley, which is diagonal from Diagon Alley.
- When Crouch-as-Moody is teaching the Defense vs Dark Arts, he mentions that the students "get your wands out and point them at me and say the words, and I doubt I'd get so much as a nosebleed" in reference to 'Avada Kedavra'... because the will to kill isn't there; Similarly, when the Moodyganger is showed using the Cruciatus curse on the spider later in the book (or when HP uses it on a Death eater in the final book) it is both through the words and a torturous intent.
- Also in Goblet o'Fire, when HP uses Accio Firebolt, it does not bring any nearby Firebolt, nor does it bring every Firebolt nearby, but specifically Harry's personal Firebolt that he was concentrating on.
- When HP uses Sectumsempra in the Half-blood Prince, he knows the extremely simple jist of the spell - It's "For Enemies". Being able to concentrate, using the wand as a focus, HP directs the spell towards his enemy, not knowing what will happen. It's the magical version of "Stick 'em with the pointy end".
- The closest we get to magic metaphysics is the discussion surrounding the Patronus charm. Lupin says directly, "With an incantation, which will work only if you are concentrating, with all your might, on a single, very happy memory". There are direct discussion referencing how it has to do with mental state, showing HP eventually succumbing after a dementor assault; it also may have something to do with inner morality, as the only Death Eater to show the use of one is Snape. The incantation for the patronas may be incorporeal or corporeal, but it has to do with the mental state, not the actual incantation.
We also see that the incantations don't need to be spoken, which is shown through both Dumbledore, Snape and Voldemort using silent magic. The Trio also does this in the movies, with stupefy, but I cannot recall them using silent magic in the books offhand. Dumbledore, Lily, & Snape are shown to use it and most cases of underage magic happen due to wandless (and therefore focus-less) magic. On one hand, those with training and discipline can direct the "magic energies" without needing a verbal or material focus; while those with the magical gift can do "things" without any control whatsoever.
Leaning and performing magic may very well may be like the transference of ideas via text - There are things that you cannot possibly know or experience, that you could learn once you understand a language. Once you can understand a language, you are no longer limited to just your own experiences, but to those who have written in that language. Later on, once you really internalize it, you can start thinking in that language for some, or all, things. Someone can sing "Voulez-vous coucher avec moi?", but not understand what they're saying. Later they can learn to read French, so they understand what they sung, but still not converse in French in an useful manner, or without a dictionary in hand to interpret. Later still they can learn the full language, can think and converse in that language, understand completely foreign viewpoints and experiences.
(If you feel you've seen this before, I stole my own answer to THIS question)
