A general term for those who use magecraft, regardless of nationality or field of study.
They don’t get along well with the Church, and the two groups are constantly feuding behind the scenes.
However, they have pretty much nothing to do with the plot of Tsukihime. If you want more details, please refer to Kara no Kyoukai.

Tsukihime Data Collection: Tsukihime Dictionary

Magi [singular: Magus] are practitioners of techniques for the artificial reenactment of Mysteries and Miracles, known as magecraft. Magi are born with a nerve-like system, called magic circuit throughout their body, which is employed to convert life force into magical energy in order to wield magecraft. To be more specific, magecraft is the ability to reenact Mysteries and Miracles that could be replicated by human civilization of the era. On the other hand, Magic is the ability to enact Mysteries and Miracles that are impossible for human civilization to ever achieve through any method or means. In the present era, where almost anything is possible through science and human cooperation, most magecraft has ceased to be Magic, and supposedly a mere five Magic remain today.

Fate/Zero Animation Guide I: Glossary of the holy grail war I

Wizards. In Fate/EXTRA “Wizard” refers to A-class hackers.
Compared to your average hacker, wizards posses special abilities and can dive into the cyber world at a soul level.
Ordinary hackers fall into two categories:

  • Visual Perception Digitization (only visual)
    (This is done through a simple spinal insert interface connection.
    Information is directly streamed into the brain, so they avoid the trouble of transforming each type of information into a digital signal. Favors input only.)

  • Visual and Touch Perception Digitization.
    (This is done through a wired connection from the nerves to the net. It allows a hacker to project the basic components of their being onto the net. Input/Output capable, and more powerful.
    However, the user is unable to project a human form in the cyber world.
    This is at best a simple projection, and as such is not recognized as a life form. Their existing circuit is read from their body, and they just manipulate that projection. They are unable to change or revise their circuit.)

However, wizards are capable of far more than that, they can digitize and spiritize their consciousness and physical body, creating a copy that is sent inside [the cyber world].
It is the materialization of soul inside the cyber world.
As for cyber bodies in SE.RA.PH, they refer to the spirit form of three components: body, mind, and soul.

Hacking knowledge and skills are not particularly important for wizards, physical ability and mental prowess are far more useful.
Having magic circuits (also known as imaginary nerves) in your body, and the ability to clearly conceptualize your digital self is essential to help translate the true essence of yourself accurately into a digital image in the cyber world.
The way of conceptualizing one’s digital image differs from wizard to wizard.
There are a variety of methods; some use computer processing,
others use meditation to give birth to their image,
and some even take drugs to escape from reality.
In present day 2030 Viritual Spiritron hacking has become possible through an external interface, but magus-class hackers remain a rarity.
Due to magus having human rights, even the West European Plutocracy couldn’t monopolize them. Aside from those under the control and management of the United Nations, there remains a never ending list of countries who try to harbor free magus behind closed doors.

The definition of life is considered to be “the ability to change the environment you live in through your own actions and of your own accord.”
Since wizards can intervene in the cyber world through their souls, Moon Cell accepts them as living, and thus their advance into the cyber world is possible.
Pioneering, expanding to new frontiers is, in a way, the greatest meaning and sin of mankind.
To wizards who posses this ability, it’s not too much to say that the cyber world is like a second reality.
A physical body is merely a “home” of sorts to a wizard.
If the physical body of a wizard on Earth were to die, it would simply mean that they would not be able to return “home,” and they would remain in the cyber frontier. As long as their existence in the cyber world is not destroyed, their soul would not disappear.
(However, the empty shell that is a Wizard’s real physical body protects them from change in the cyber world, and the loss of it will quite easily and unavoidably transform the structure of their intelligence, as well as their sense of values.)
To a wizard, the only true death is the death of their cyber existence.
Only after their souls have burned out and crumbled away will they become scrap that is no longer able to be revived.

Translator’s Notes
  1. ^ In Fate/EXTRA Material, this entry appears under the あ section (instead of ま).
Fate/EXTRA Material: Encyclopedia of Fate/EXTRA

One capable of wielding magecraft. Mages pursue mysteries, and strive to reach the Root.

MB Back Alley Alliance Nightmare: Encyclopedia of Rojiura Nightmare

Wizards are the A-Rank hackers in the Fate/EX series. Compared to other hackers, they possess an unusually high level of talent and gift, and can send their very souls into SE.RA.PH.
They transform their souls, minds, and bodies not into digital information, but into spiritrons. With their engineering skills, they can then use these avatars to navigate cyberspace.
In other words, they are able to fashion “physical” shells, called Cyber Frames, within the digital world. Cyber Frames being the combined spiritron embodiment of body, mind, and spirit in SE.RA.PH.
While knowledge and skill are both obviously important to hackers, what’s most important is talent of both mind and body.
To be a true hacker, one must have the rare ability to clearly imagine the nature of one’s body virtually as a sequence of numbers, utilizing a set of “virtual nerves” known as “magical circuits.”

Sentient beings within the digital world, whether they have truly physical bodies are not, can still be considered “alive,” in that they can affect change in their environments according to their own will and conditions.
The Moon Cell recognizes beings with this ability, and allows them to explore its virtual space, pioneering new frontiers — as has been mankind’s greatest purpose, and greatest sin.
For Wizards who have access to these possibilities, one could even say that cyberspace is a second reality to them. A Wizard’s original body, back on Earth, is merely its “house.” Even if one’s body on Earth should perish, they will be only be unable to return to that house, and nothing more. They can remain in the new frontier of cyberspace, and as long as they are not extinguished there, their soul will continue to exist.
(However, Wizards who do lose their physical bodies also lose those bodies’ natural resistance to change. While they can continue to exist, their thought processes and value structures easily bend and warp, and a change of nature becomes as unavoidable.)
For a Wizard, “death” is the death of his or her digital brain. The moment his or her soul burns out in the cyber world, they irreversibly turn to dust.
The Main Character is a Wizard who now lacks a body on Earth, due to past events explained in a previous game, Fate/EXTRA.

Fate/EXTELLA material: Encyclopedia of Fate EXTELLA