HP magic works not with the physical world, but rather with concepts. Two people casting stupefy might use wildly different paths, but both lead to the same effect, same concept – lack of consciousness. Avada Kedavra uses the concept of absolute death, which may take form of heart stopping, brain damage or sudden ceasing of all nerve impulses. Crucio uses the concept of suffering – no two Crusios are alike.
In order to cast a spell, a wizard reaches toward the Platonian world of ideas (also known as the Root or Akasha) and creates an imperfect copy using magic.
Accidental magic is possible because as a child, person lacks better comprehension, and rather than using a specific concept, he just crafts a less than complete copy of random cluster of ideas. Controlling accidental magic is inherently impossible (see contradictory concepts).
"Power" of the wizard hinges not on some abstract "magical core" (see sorcerers) or ability to use magic without spells, but rather on one's comprehension of the concept. While it is impossible for a human mind to truly understand one, a better understanding causes a more comprehensive rewriting of reality.
A wand embodies the concept of Tool. Ergo, a better wandmaker with a better understanding of said concept is capable of crafting better wands. It also means that wandless magic is possible, but similar to trying to hammering in a nail with your bare hands – slow, uncomfortable and inefficient.
Casting spells on a human, especially a wizard, is a complicated process, since each sentient being is capable of comprehending (and thus embodies) a cluster of concepts. In order for the casting to be successful the concept (not the spell) needs to be understood by both parties and be as simple as possible. One example of such concept is death – instinctively understood by any human being, but incomprehensible to both dementors and phoenixes (see immortals).