Magic is the art of making something happen by unusual or paranomal means, by conducting Mana through something, most commonly a living object. Many different intelligent species can cast magic, most notably Aefians.
Magic is done by siphoning mana from the Soul, through the brain and into the heart, where the brain through a series of electrical impulses moves it to the skin, near the area it is intended to be cast. Most commonly, this are the hands or eyes, where a simple point or gestre, along with a memory of the spell, can cast it. This is not the only case, however.
The methods of casting magic are as varies as the poeople casting them or the spells used, but the most general consensus is by conduit, memory and thought, array, symbol, word or even movement. Often with mages there will be a combination of all of these, and the lines between each are fuzzy.
Conduit relies on some sort of midway point to convert the raw mana into a spell, sometimes coupled with a word. Examples include Harry Potter.
Memory and thought rely simply on a thought, gesture or movement as well as memory on how the spell works. Examples include most of the Imaginaughts, as well as the Elder Scrolls V: Skyrim
Array works around using circles and lines to direct mana flow. Examples include Fullmetal Alchemist.
Word requires some sort of noise, word or grunt. Examples include Dragon Ball and, again, Harry Potter.
Movement is based around cirtain movements to direct the flow of mana, like a literal array. Examples include Naruto.