Magician: A magician (魔術師 Majutsu-shi lit. Magic User) is a person trained to use magic. While any human being can use magic as long as they follow the proper steps and precautions, magicians are extensively trained and suffer less risks if something goes wrong during the process.
Magic: Magic is a complex system of arts, disciplines, and traditions beyond the knowledge of science that allows its users to manipulate and distort the world around them. Unlike esper powers, magic usually requires some sort of previous preparation; be it the making of a temple and a ritual or the construction of a Spiritual Item or Symbolic Weapon. However, unlike espers, which are restricted to one power, a magician can wield several types of magic. There's two broad categories of magic: Idol Theory or Imitative Magic, in which an imitation of a whole or parts of a subject, the source, gains properties and attributes that are inherent to that source (which could be an item, weapon or story), e.g: a replica of a magic sword gaining properties of the original; and Infection Magic, in which parts of a subject (items or even beings) are used to remotely affect or attack it, e.g: an effigy used to attack a person through one of their hairs.
Mana: Mana is the energy that allows magicians to cast magic, and is present in all forms of life. There's also natural mana sources that some magicians and Spiritual Items may use in the form of ley lines. One way of using mana is through the refinement of the user's inner lifeforce, similar as to how crude oil is refined into other substances. As the source of mana is one's lifeforce there's a limit as to how much mana a magician can produce. Once this limit is reached their own body will refuse to continue the process out of self-preservation. Similarly, it's speculated that immortal beings like the mythical vampires might possess unlimited mana due to their inexhaustible lifeforce. Due to their training magicians have larger amounts of mana than the average person, which allows them to tell if someone is a magic user just by standing close to them. Additionally, each discipline of magic (Norse, Shinto, Christian, Hindu, etc) requires different kinds of mana, which sometimes allows experienced magicians to perceive which discipline a spell or spiritual item belongs to.
Telesma: Telesma is the power dwelling in the Phase known as Heaven and the source power for Angels. Some items or magicians use Telesma to achieve great power, since the potency of Telesma can't be compared to normal mana. The human body cannot handle a large amount of telesma due to a difference of affinity, not a matter of potency. If a human does forcefully absorb a large amount of Telesma it will result in harmful effects in their body.
Spiritual Item/Symbolic Weapon: A special tool prepared by the magician in advance to cast magic more efficiently and conveniently, normally using Idol Theory to grant the item/weapon power. As long as the item in question is similar in shape or function it will gain some of the original's properties, like an umbrella with a knife on the end standing in for a spear, but it will obviously gain less than a more faithful replica would receive. There's some items, especially those of divine origin like the Longinus Spear, the Holy Grail or divine weapons, that can't be accurately replicated through normal human techniques.
Saint: Saints are a powerful and rare existence in the Magic Side, with less than 20 of them existing in the world. Their power is based on Idol Theory: Humans are said to be made in the image of God, and Saints are said to be a higher replica that can bring out more strength, having a body which is closer to the Son of God due to being naturally born with God's blessing. Traits that most Saints have include superhuman physical strength, incredible speed, the capacity to channel a larger amount of mana compared to normal magicians and superhuman senses (for example Kanzaki Kaori has 8.0 visual acuity and can easily tell people apart at 600m, while Acqua of the Back can easily hear a whisper at 10m). They are also blessed with success and have naturally high luck. For example, Kanzaki was stated to be able to easily win over people's hearts without trying and has survived multiple assassination attempts through pure luck, such as a bullet deviating mid-path for no reason or a bomb exploding right next to her and failing to even scratch her. However, their same unique traits also make them more vulnerable to attacks that resemble how the Son of God was killed (being stabbed, a crown of Thorns, and the crucifix), though this weakness normally is inconsequential since most magicians don't have the power to exploit these weaknesses, and most Saints are aware of this problem and take measures to avoid being affected. Due to their power, Saints are capable of taking down an entire city or magical organization on their own, and they are considered comparable to nuclear weapons in terms of their importance in the Magic Side.
Magic God: Magic God (魔神 Majin) is the term used to refer to a magician that has mastered magic to the point of being able to bend all things in the world without exception through magic, becoming a being that transcends normal magic and steps into the realm of God. As such all possibilities are open for their power, like turning the sky green, resurrecting people, making 2+2 equal 5, controlling time or even destroying the universe. But that produces an omnipotence paradox, because all possibilities also include negative possibilities like losing against a child in a game of rock-paper-scissors. This paradox normally restrains a Magic God's power to work only 50% of the time, but a Magic God can eventually overcome this weakness. A Magic God's knowledge of magic is so great they are likely to be able to copy any spell they see being used, even if it's just once. The process of becoming a Magic God has not been described in detail and it seems to differ from Magic God to Magic God, but from the ones who have appeared in the series it seems to require a ritual of some kind.
Phase: A Phase is a layer applied over the world, in this context referring to the entire universe or reality itself, which acts like a veil or filter, and changes how the world appears to be. Several layers exist in the world, with many of them having been created by religions through unknown means. The known phases in the world include Heaven, Hell, the Underworld, the Pure Land, Yomi, the Abyss, Olympus, the Fairy Island, Nirai Kanai, Asgard, the First Sun, Takama-ga-hara and Amaravati. Beyond all filters there's a world of science untouched by religion or magic, referred to as the Pure World. Magic Gods have been shown to be capable of warping reality as they see fit by creating, erasing or manipulating these Phases. Some weaker magicians and magic items are also able to affect Phases in some way.
Original Grimoire: A grimoire is a book containing esoteric knowledge about magic and powerful spells. Only a few experts can read grimoires, as the knowledge inside a true grimoire is of a different world, and contradicts common sense, a knowledge that is far too alien to be safely comprehensible by any normal human, whose minds and bodies will suffer from its "toxin" if they try to read it. That "toxin" is the result of the reader’s mind rejecting such a great discrepancy from their common knowledge and wisdom. Even those few experts are threatened by grimoires, as the knowledge that a grimoire possesses is "poisonous" in the sense that it is too "pure" for the human mind to handle and must be "watered down" - i.e. written in such a way that many of its meanings are obscured—in order for people to be able to read it without their minds being destroyed or being led into insanity, plus a variety of other harmful effects. Grimoires are practically "eternal" in the sense that they cannot be destroyed with current human methods (such as burning them) and can only be sealed away, as they use ambient magic like leylines to protect themselves. Grimoires possess some degree of sentiency, as they desire and act to spread the knowledge they contain to people, and may even turn against their users if they feel that they are threatening their chances of spreading their contents.
Esper: Esper (能力者 Nōryokusha, lit. Ability User) is a term used to refer to a person who emits AIM. It's generally used to refer to people who have gone through Academy City's Power Curriculum Program and artificially gained an ability, though natural espers called Gemstones also exist. Unlike magicians, espers only have a single power, but they usually do not need any previous preparations to use it. Additionally, despite only having one power many espers come up with several aplications for it. Esper powers are clasified into two groups: Psychokinesis, which lists all powers that allow the user to physically influence the world around them, e.g: Telekinesis or Pyrokinesis; and Extra Sensory Perception, which lists all powers that involve the reception of information not gained through the recognized physical senses but instead use supernatural senses, e.g: Clairvoyance or Thoughtography.
AIM: An Involuntary Movement, better known as AIM, is a term used to refer to the phenomenon in which an esper involuntarily produces a weak invisible energy around themselves, while the AIM Diffusion Field is the energy field that surrounds the esper that is made up of the aggregation of that energy. The AIM Diffusion Field affects the esper's surroundings in microscopic ways depending on their power. For example, pyrokinetics heat up their surroundings, telekinetics create pressure, and electromasters release electromagnetic fields. The AIM Diffusion Field is normally only detectable through the use of machines, and is unique to each esper.
Personal Reality: Personal Reality is the source from which all esper powers and phenomena are brought into the real world. It's the esper's distorted personal vision of the world, thus being unique to each individual and irrevocably linked to their ability. It allows the user to utterly ignore the Uncertainty Principle, and with regards to the Schrödinger's cat thought experiment, would allow them to choose the possibility they want, or more specifically, distort a micro world using different laws, or more accurately, control their microscopic observations of the world and use them to cause effects on a macroscopic scale.
Level: Espers in Academy City are classified in a Level system depending on the strength of their powers. The system goes from Level 0 (Person with No Powers, any power they have is too weak to be of any use or they are unable to consciously control it) to Level 5 (Person with Super Powers, powers of an extent that their wielder can fight alone against a military force on equal or superior terms). A Level 6 (Person with Absolute Powers, the user is considered to surpass humanity and enter the realm of God) has been theorized to be possible, but so far nobody has been able to reach it.
Magician-Esper hybrid: While it's true that anyone can learn how to perform magic, espers are usually prevented and advised against doing so. Several collaboration experiments between Academy City and the Magical Side have proven that any esper that uses magic will suffer severe damage, as the two power formats are incompatible. The body part that is damaged is seemingly random, and as such one esper could be able to cast magic a few times without any fatal damage or immediately have their heart or brain severely damaged by their first spell, something that has drawn comparisons to playing Russian Roulette. The only successful magician-esper hybrid is Tsuchimikado Motoharu, as his esper power repairs the damage his body suffers with each spell. However, even his ability to recover is limited and could fail at any moment, so he only uses magic when there's no other choice left.