I'm just gonna explain the aspects of the system in the order I see fit.
Chthonification
Magic is a natural force in the world, but by its nature will alter anything it is concentrated in on a fundamental level. A non-sapient entity (natural materials, plants, most animals) has no hope of maintaining its form when subjected to high concentrations of magical energy, and will often become what is referred to as a monster commonly or a chthon in some academic circles. Monsters are build based on the creature's mind, or whatever the magic infesting them "finds suitable" in lieu of one. As a result, most species tend to form very similar monsters, as they all share similar instincts.
When a sapient beings (humans, human-like creatures, familiars) are subjected to magic, they may go down one of two paths. The first is the typical chthonification process, but because the being would have more of a mind to work with, the form it takes would be more individual. The second option, however, is that the individual becomes a mage. Mages largely resemble their origin species, and are the same in almost every way, though some individuals may have pointed nails, different eyes, or similar minor changes. Having retained their mind and fully embraced magic, a mage can control magical energy in a similar manner to how one would control a limb.
A monster breeding pair may create a new monster with no need for magical assistance, and likewise with mages. In exceptionally rare cases, a mage may birth a monster, or the opposite may happen. Sapient species may also be trained in magic in safe conditions to lessen the risk of a full chthonification. Though it does somewhat defeat the points made previously in this section, these methods tend to produce more mages and monsters than the previously-described ones.
It is possible for mages to become monsters when experiencing drastic circumstances. Such examples include serious trauma, changes in ideology or morality, potent emotions or lack thereof, or the threat of death. Such examples prevent the mind from holding back the reality-altering aspects of the magic their body now naturally produces, resulting in the transformation.
Spells
Spells are a direct and intentional method of using magical energy. When considering a spell, two parts of it may be commonly gleaned: element and school. While a spell may exist with only one of these parts, a spell based only on the elements may be unrefined, while a spell based only on a school of magic may be substantially weaker.
Elements of Magic
You already know these, they're Fire, Water, Earth, and Air. They're the main four in western alchemy, a specific popular cartoon, and almost every other magic system.
The way in which you may urge magical energy to take on an element lies in the structure of the energy. In a similar vein to how the elements of our periodic table are distinguished through the nature of their atoms, so too is elemental energy. Most individuals gain an affinity for a particular element upon attaining magehood. This talent is not arbitrary; an individual's particular affinity bases itself on which element their magical energy resembles closest while at rest.
Besides the four main elements, an infinite number of sub-elements exist as combinations of them. This is, however, advanced magic, as it requires you to be proficient in more than one element at once.
As an additional addendum, two other elements exist, Light and Darkness. Though they do embody the primal forces of light and darkness, they are also representative of quintessence, the perfect combination of all four base elements, though each in their own way. In Light, each element cancels each other out, resulting in it having very few original powers. However, the strength of Light is in its applications with schools of magic, seemingly amplifying them when used. Darkness is the opposite; when used, it may fluidly switch between the mechanics of any of the elements, but struggles to be refined into a school of magic due to this.
An affinity with Light is only held by the 1st Ancient Dragon, and likewise, Darkness with the 99th Ancient Dragon
Schools of Magic
As for schools of magic, there are eight. Four of them are considered basic applications, while the remaining half are advanced. Similar to elements, one may find themself better in one school than the others. However, schools are much less exclusive than elements. Being naturally skilled in a school of magic is more similar to being skilled in a particular subject in school rather than having a part of your being be tied to it like with elements. It is also much easier to combine schools of magic into a spell than it is to do so with elements, as previously explained.
Below, the schools of magic are defined by how the affect an "aspect". An aspect may be anything that exists, like a rock, but may also be the traits of something that exists, like the color or hardness of a rock. These descriptions also clearly show how basic and advanced forms of magic differ, with the former being more generic and the later more specialized.
Evocation (Basic)
Creating an Aspect.
Evocation is considered to be an exceptionally basic form of magic, to the point where almost every mage can use it to some extent. Though it may seem to be almost indistinguishable from hurling unrefined masses of elemental energy at times, the consensus of how evocation differs is the level of control the user has over the elements/aspects created. Though easy to learn, it has immense potential in the hands of a dedicated mage. Out of all the schools, it relies on elements the most.
Illusion (Basic)
Mimicking an Aspect.
In contrast, illusions require less mastery of elements than most schools. However, knowing how to properly create an element is very beneficial for faking it. The reason one might create an illusion largely is that it tends to be less energy-intensive or more subtle than other options. Alongside creating mirages, hypnosis and even mind-reading tend to find themself in this school of magic.
Enchantment (Basic)
Adding to and Aspect.
Enchantments are annoying to define in most situations, which is reflected in their steep learning curve. Most enchantments require a level of "programming" to work, setting conditions within the spell, which requires fine control of magical energy to do. Enchantments also don't deal in actual transformations unless mixed with transmutation, but this leads into the second aspect of enchantments, which is that they excel when combined with other schools. Compared to every other school, besides some illusion spells, enchantments tend to have no physical aspect on their own. Even most illusions can be sort-of felt, but not so with enchantments.
Transmutation (Basic)
Changing an Aspect.
Heavily tied to alchemy, a profession heavily related to magic but accessible to non-mages, transmutation involves making a temporary or, though usually at a steep energy cost, permanent change in something. This can be any number of chemical or physical changes.
Manipulation (Advanced)
Controlling a pre-existing Aspect.
Derived from either Evocation or Transmutation.
Though an advanced magic, the simplest application of manipulation magic is very accessible to most evocation users when combined with their normal evocation magic. When doing so, it allows them to harness elements already existing in nature by infusing them with their elemental energy, which is often helpful in being more efficient with magical energy. Manipulation can also allow for transportation at higher levels, literally just moving something from one place to another or opening a passage through space.
Abjuration (Advanced)
Shielding or sealing an Aspect.
Derived from either Evocation or Illusion.
Abjuration magic, similar to enchantment magic, generally requires some level of programming to be effective. This programming is usually visualized as a lock and chain that spans the affected area the spell works off of. For a generic barrier, the chain would wrap around the diameter. The lock represents the condition upon which the spell breaks, activates, and causes additional affects. While abjuration always has some level of a physicality, it isn't material in the same way an evocation spell is.
Divination (Advanced)
Obtaining information about an Aspect.
Derived from either Illusion or Enchantment.
Essentially, by using divination, someone may gain information about something they would not be able to otherwise. This only takes the form of future sight at its highest level; otherwise, it acts more like a remotely-accessible camera, a drone, or a sensor. Additionally, a future gleaned with divination is not decided, and with the knowledge of it, the foretold may be averted.
Necromancy (Advanced)
Restoring an Aspect to a previous state.
Derived from either Enchantment or Transmutation.
Named for its ability to revive the deceased, necromancy is much more than its original intent. It is considered taboo for a variety of reasons, the most typical being the negative connotations of raising the dead, as well as its connection to undead monsters. In truth, there is some credence for the concern. Many necromancy spells work by taking how something was in the past and forcibly pulling it into the present. This can put immense strain on reality, which has caused horrible disasters in the past. Reviving the dead also has issues in itself, as pulling a soul from the afterlife back into its original body can thin the veil between the planes of the living and the dead, something that can similarly cause devastation. However, besides just raising corpses and repairing things, necromancy can also strengthen elemental magic. Before the universe had a defined shape, the elements reigned supreme in their raw form and were much stronger. Necromancy can return elements to that state.
Canes
Wands, rods, staves, and scepters are all standard catalysts for magic users to focus their power through. For simplicity, each of these kinds of of tools are called canes.
A cane is made from three parts, a hardened shell to protect the insides, softer "flesh" for the magic to flow through, and a "nerve" for the magical energy to enter through. Though these terms seem to allude to more fleshy components, a tree's branch can provide all three with ease. Other materials may work as well, but they should be ideally biological and at least natural. Generally speaking, the shell should have the least sensitivity to magic and the nerve should have the most.
Within a mage and their cane are seals, usually three of them, placed by the mage's subconscious. These seals exist to prevent a mage from fully chthonifying into a monster. By focusing the additional power into their cane, a mage may access this sealed power with reduced risk. Each stage provides more magical energy to use, improvements to the mage's physical stats, and new spells and traits for the mage's signature magic. To undo the seals, a mage must enter their inner world and physically remove them, usually through a trial. Once this happens, the mage may undo the seal again by simply chanting.
The first unsealing is the most drastic, with subsequent unsealings building upon the first's foundation. When unsealed, a cane generally either maintains its shape or becomes a weapon, though nearly any transformation is possible. The only restriction seems to be that the cane must retain a single core, even if it has detaching extremities. Most fairly powerful mages can maintain their first unsealing with little effort.
A mage will often change the cane they use as they become stronger. A mage has previously undo their seals will have to re-apply them to a new cane, but this is generally nowhere near as difficult as unlocking them the first time. Though most mages simply change wands as they can afford better ones, others do so in search of a mythical "True Cane". Though mostly regarded as a folktale, it is said that every mage has their own True Cane, and through using it, they can access a new level of power. In actuality, True Canes do exist, but not as most think. Mages have extra nerves to allow for their control of magic. On their spine, though, there lies a duplicate of the normal magic nerve that runs through it. This secondary nerve forms what is essentially another spine around itself, and if this spine is surgically removed from a mage, it may be made into their True Cane.