Magic is an often used term in fiction. It is the concept of affecting the universe through supernatural methods and breaking the physical laws.
Magic is often used as to explain the unexplainable. The method of applying and using magic varies greatly from story to story. It's almost always limited to only a special group of people and the use of certain magical words to activate it. The need of magical devices (generally staffs or wands) isn't uncommon.
In the Nanohaverse magic is like advanced technology. The background explains that an advanced civilization threw out the conventional weapons and turned to magic instead. Since then the overseeing the magic and the use of mages by the them became common.
Of course as a Magical girl show the Nanohaverse has the Intelligent Devices. They are small devices which can talk and assist the mage.It stores the mage's special outfit(Barrier Jacket) and magic staff(or other weapon). Compared to a regular magical girl show the designs and functions of the Intelligent Devices are more technical and they give off the feeling of advanced technology instead of elaborate or mysterious. It's admittedly becaus the author, Masaki Tsuzuki is a fan of Gundam and mechas in general.
It's a running gag to call the magical girls(especially Nanoha) as human-sized Gundams. Certainly, their firepower matches their fame and Nanoha even gained the title of White Devil(the nickname of the original Gundam Mobile Suit and its pilot, Amuro Ray).
The basics of the magic of Negima follows the classical concepts known in fiction and RPGs. Although the basis quickly branches into two by the introduction of mana and ki. Mana is the energy of the surroundings for power. This is the classical art of magic which the Western mage apply. On the other hand the Eastern magic concentrates more on Ki, the energy produced inside the body. The effects are similar though the methods usually differ. Most western mage rely on rituals and long incantations to cast spells. The Eastern mages use charms and talismans to use magic also they often undergo martial arts training and fight directly.
As the Western mages are often inept in direct combat they developed the Pactio. The Pactio is a contract between Magister(master) and Minister/Ministra(servant). The Magister/Magistra offers portion of his/her magic power to the Minister/Ministra and in exchange the later protects the former. Every successful Pactio also give a magical Articraft to the Minister/Ministra which possess a special ability. The Articraft reflects the character and personality of the Minister/Ministra or the skill which he/she's the best at. Although later in the story the protagonist, Negi Springfield becomes fine combatant on his own, making the Pactios similar to a regular power-up in the story.
In terms of spells Negima sports wide variety: ranging from various elemental attacks and shields to summons, illusions, mind-reading or even pocket-worlds where the time flows differently. Also after the first few bunch of chapter it became natural that everyone uses magic power for physical enchantments like super-strength or speed, making any high-class mage superhuman.
Thaumaturgy or Magecraft is extremely detailed and full of technobabble-like explanations. It makes magic look more like a secret branch of science. Secret because as in many fictions the existence of magic is a secret from the normal people. The first requirement for a mage to possess Magic Circuits. One individual can possess more than one of them and their quality and quantity determines the talent and power of the mage. The mages are going through rigorous training because casting of the spells can be really exhausting or even painful.
The society of mages is gathered into one big organization, the Mage's Association. It's a corrupted and secretive organization which views the advancement of magic as its ultimate objective. Although keeping up the masquerade is its top priority and the Assotiation has great influence in many governments to clear up any mess the mages cause. As long as a mage doesn't expose the magic, the Association usually has no qualms about what he or she does. Even if the said mage is a mass murderer. There are two other large organizations and the three is in a constant though regulated war for power. one of the two is the Dead Apostoles. It's group of vampires lead by the 27 Dead Apostole Ancestors. The other faction is the Church, with the objective to clear of the magic and impure creatures. As such the Church is the natural enemy of the vampires and the Association. All three factions have their own reasons to keep the world of magic secret from the common people.
The world of supernatural has many secret powers and forces. The True Magics are one of those. They're magics with the power of true miracle. They're unexplainable for even the mages. In the present timeline only few of those exist but the mages consider magecraft as a way to recreate those miracles.
The magic of ZnT generally works in the classical way. There are four elements: Air, Earth,Fire,Water. The more experienced mages can control more than one element. A triangle class mage for example can control three elements while a square class can all four of them. Although there's a special element, the element of void which possesses mysterious powers and gathers its powers from the emotions. How convinient to grant it to the heroine who has the worse temper...
The mages generally possess a familiar which they summon from other worlds. And this is where the story starts when someone accidentally summon a human for familiar.
Later there are more complications and even some hints of conspiracy but despite that ZnT's magic is one of the simplest of the lot.
In Shana, the source of magic is the power of existence. It's a power which everything yields but especially living beings, humans.Once this power is drained from something, the said object or being disappears without a trace and no one will remember it. It clears out its existence completely. The Guze no Tomogara tries to seize power by forcefully draining this powerwhile the Lords of Guze are trying to prevent that. They are deploying the Flame Hazes to slay any Tomogara. To prevent fatal collapses in the world the Lords of Guze use torches, temporal copies of people to replace the ones fallen until the world adjust to the change better.
The magic works through unrestricted spells and various flames. Fuuzetsu has to be a honorable mention of the spells. It freezes time in an area and only Flame Hazes, Tomogara and some other special beings can move inside it. The former two is usually responsible for the Fuuzetsus also. They set that up generally in every battle. Other than magic abilities there are many magic items, Treasure Items. They possess great powers and when a Torch somehow has one inside them they're called as Mistes. Both Tomogara and Flame Hazes are searching for those items and the Mistes.
In TRS the party travels between dimensions to reach their goal. In each dimension the rules of magic can differ and be unique to that world alone. There are only a few general rules for magic. For example no matter how powerful the mage is they can't resurrect people. As the introduction suggests powerful mages can even cross the various dimensions with their abilities. Some of them can foresee the future to varying degrees and they can even travel in time or clone people with magic. The group's main benefactor, Ichihara Yuuko was one of them before the story and she still possesses great powers. More details are hard to find as CLAMP is rumored to be notorious for not explaining the workings of magic much. At least if the story has little need for it.
In Umineko no Naku Koro ni there are witches who use magic on a regular basis. They are incredibly powerful and generally cruel(or appear to be). They each have their own specialty where no one else can surpass them. The power of magic brings miracles but it depends heavily on the power of belief. The magic can work via any possible methods ranging from written runes to simple weave of hand. While it seems limitless it has a few limitations. Generally it's impossible to resurrect the dead or permanently restore which was once broken. The only exception is for the one who has the title of Endless Witch whose magic is permanent and never expires. Also magic is ineffective on someone who absolutely rejects its existence(like Battler, it's one of the major plot points of the story).
In Slayers there are four different type of magic dark, white, shamanistic, and holy. Dark magic is a offensive type of spell. Its powers come from pain and misery of the world and more powerful spells come form Mazoku. White magic is heal, protect, and send spirits to the afterlife. Shamanistic magic come from nature and the astral plane. Shamanistic spells are more useful in fighting Mazoku because the spells can hit there astral body. Holy magic have be lost since the Koma War. Its powers come from joy and happiness of the world and more powerful spells come from Shinzoku.