Magic
Magic is defined as utilizing mana to initiate some kind of effect. Naturally, this is a somewhat vague description, but for the purposes of defining magic it is the only valid one. After all, the vastness and complexity of the spells that have developed in the time leading up to the modern era is incredibly difficult to summarize.
All mana originates from the limitless Origin, where it is colorless, "raw" mana, magical energy in a pure form. When mana leaves the origin and becomes the mana supply of a mage, its quality and quantity are defined by blood. While there are ways to compensate for having low amounts of mana or low-quality mana(or both), traditionalist magi tend to feel that these methods are slipshod shortcuts at best. More modernist magi may be more open to those who use such tactics. When mana becomes a mage's source of power, it is "colored" by that mage's affinity. As a result, that mage becomes best suited to performing spells within their affinity, and the mana is no longer pure. This is not to say that a mage cannot use any spells that do not belong to their affinity, but it means that they are best at spells falling within it.
Affinities are passed through bloodlines, and can be nearly anything that is not made by man. Though some affinities are extremely rare, there is no such thing as an entirely unique one. At least, almost no such thing. There are two exceptions to this rule, the enigmatic Mana affinity, possessed only by Nakamura Kiyomi, and Death, possessed only by the Arakawa bloodline(and only in a usable form by Arakawa Yoshika, the only member of their family truly aware of the supernatural on a large scale). As previously stated, an affinity dictates what spells any given mage is best with, but is not exclusive. A mage can learn other spells, but it will be more difficult(Nakamura Kiyomi is an exception, as she can only use spells to direct mana to use it for defense and offense).
Spells themselves are an incantation or chant that allows for mana to be used to create some form of effect, be it a ball of fire, healing an injury, or anything else. The more powerful a spell, the longer the incantation. Rituals can be used to activate spells as well, and runes can be used to remotely activate spells without vocally casting them. Mentalcasting is a process that allows a spell to be cast mentally, at the cost of some power. Even the most skilled mentalcaster must say the name of a spell to use its full power. This is for magi, of course. While magi are the most wide-spread form of magic-user, Japan's onmyouji utilize a method like runes in order to cast spells, using ofuda, paper slips inscribed with various spells, in order to use magic.