What is magic? This is the question asked by every young student of magic at some point. The answer is, like all such answers, easy to explain glibly, hard to explain properly. There are additional confounding factors in the implementation of magic in our realm, which we will also discuss.
Magic is, at its heart, a way to utilise free energy of a type not native to this universe, to affect the world. Which seems simple, but is also not exactly correct. It can be shown* that magic does not directly act to increase energy in this universe, but rather acts upon it in ways that are as yet unknown to achieve the desired result. Thus magic does not create the lightning bolt, but the almost infinitely unlikely scenario of many electrons from the object of your wrath suddenly simultaneously deciding that they are not attached to it, and instead are in the air above it, can occur. The practical difference is that a magical lightning bolt is much more damaging than a regular lightning bolt, as the trauma is inflicted twice. It should even be possible to focus the loss of electrons to the nervous system, which would almost certainly cause instantaneous death, however such an incantation is beyond mortal spell casters for now. Immortals of course have much easier ways to kill someone and have never had to look into this method.
Magic leaks across Dimensional Bridges and Dungeon Gates into this universe, however once here it has to at least obey some of the laws of this universe. Thus the creation of High Magic Zones. These are defined as a sphere around a source of magic, where the magical energy (henceforth called mana, as is common), saturates the atmosphere and other matter in the region such that there is free mana available for use by biological entities† and permanent enchantments. This also makes spell casting easier and more efficient, as the mana required to cast a spell is reduced by the amount of mana needed to drain the surrounding world of mana. (Think of this as the power needed to pump oil up from the ground, a small initial investment is paid off by the resource freed.)
SPELLS
In the Other Realm, magic is pervasive, natural and has been part of evolution since life began there. As such their use of magic is instinctive, continual and as much a part of their biology as processing oxygen is for us. However, for us, a more formal organisation of magic is needed. The spells of magic are the means by which we can manipulate the world. These take the form, normally of an incantation, runes, hand gestures or a combination of these.
The incantation can technically be in any language, however to differentiate between accidental and deliberate casting, Latin is the most common language used. Other languages used for the purpose of making the spell more difficult to cast accidentally include: Ancient Greek, Ancient Hebrew, Aramaic, Old Norse and in a surprising turn of events, 1337speak.
The runes are the symbols used to represent the spells, they can be Nordic, or just invented. One school uses Cuniform Linear B, however anything that won't be used accidentally is fine. This is, for example, no one uses Greek characters, nor Cyrillic, nor even Latin. Dead languages are safe. A note must be made here on Liturgical Latin, that used by the Catholic Church. The use of this has been restricted to masses, christenings and other ceremonies. In these contexts, it is an exhortation to their god, and any magical side effects are positively beneficial to the church.
Spells are categorised into different "schools" whereby related spells are gathered. Each of these schools has an associated expertise (see below), which increases its efficacy and efficiency. Everyone who has been exposed to a high magic zone has the ability to cast spells, to a greater or lesser extent. However it is the choice of learnt spells and the associated Expertise that leads to broad categorisations such as "fire mage" or "necromancer" or "healer".
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This is further compounded by the effect of classes which can provide substantial boost, and penalties, to a certain school of magic. For example a Barbarian class would have a substantial penalty to arcane magic, but even a low level healer could cure some quite serious injuries without too much effort.
ABILITIES, AN INTRODUCTION.
Abilities are much closer to how the home realm of magic experiences the use of magic, though far more constrained and narrower in focus. These are instinctive, set piece, spell activations with no incantaion, runic or gesture based priming needed. They have rigidly defined outcomes and cannot be varied or changed. They run on a "cooldown" system, whereby it takes a certain amount of time until they can be used again, this time varies from ability to ability and with the user's affinity. Affinity is a passive bonus given to people as they specialise in an ability tree, it increases their effectiveness, lowers their cooldowns and allows the user a greater instinctive knowledge of when best to use them.
Unlike spells, which must be learnt, abilities are granted at certain levels, and are organised into trees. So swordplay is a tree, unarmed combat is a tree, movement enhancement is a tree. Some trees require specific equipment (weapons, guns, etc.), but there are many that anyone can take. First Aid, for example increases the effectiveness of first aid treatment. Someone who has specialised deeply in First Aid can perform medical miracles with only the barest of supplies, or can staunch a wound effectively with their bare hands. This is obviously an application of healing magic, however it has so many more limitations in it use, but so many fewer in its methodology, that it is definitely different to normal healing spells.
Abilities are not confined to combat or combat adjacent fields either. There are many, many crafting, research, interpersonal skills and scientific trees, which at first glance seem like a strange conjunction. Sadly the System did not take all its knowledge from one easily defined system. So we have a Magitech tree, gunsmithing, crisis counselling, sword-smithing, enchanting, cursing, microelectronics, quantum coherence in Bose-Einstein condensates (a very specific tree), Legume farming (highland) and Legume farming (lowland), and a host of others in every field one can think of.
For the non-combat trees, there exists a secondary experience requirement, called expertise. Expertise is gained by the use of these skills, and helps to advance them.
ARBITRATION.
Occasionally the System is unclear on a certain interaction between abilities, spells or monsters. When this happens the matter is referred to arbitration. The system can manifest a sentience which will rule on the interaction. While the sentience is based on a human, it's been carefully altered to make it impartial. So do not expect an arbitration to go your way just because you're human. In nearly all cases it's better to try and avoid arbitration where possible. The only field where it's regularly encountered is research, where there exists a whole sub-field of System, research dedicated to trying to use the sentience to extract more information about the under-lying rules of the System. Thus far the sentience has proven intractable on all subjects outside the scope of the ruling it has been manifested to rule on.
* Lee at al, The non-conservation of entropy in magical computation and the implications for predictive modelling 2142
† This does not only apply to sentient lifeforms. The oft cited example of this occurring is the appearance of cyrptids of various types around dungeon gates of greater than category five.