Magic as Numinous Power
What exactly is magic? In fantasy, it's usually a form of power that is both numinous and supernatural. Supernatural here means "not known to nature" and "outside our current understanding of nature" (so not necessarily unknowable in the way some conceptions of god is unknowable, but unknown as far as our current scope of knowledge is concerned).
But what really sets magic apart from other forms of power like technology is its numinous quality: that is, magic suggests there are forces at work more grand, mysterious, and fantastical than our everyday experience: a cosmic music in the air, a vast gulf of mystery in the stars, a secret depth within ourselves to plumb... that inspires in us a sense of wonder.
In game terms, magic is the result of working a spell, miracle, or psionic power—each a type of supernatural act that allows spellcasters to do impossible things. In general, the rules will use the word "spell" to refer to any of these acts.
Sorcery is magic used by mages and is generally understood to be magic “ex mundo,” meaning, magic "from the world." Mages produce magical effects by casting spells, which are encoded in complex, mystical formulae that can be learned through academic study.
What are Spells?
Each spell is a set of instructions—written as pages of prose or poetry, with the linguistic complexity of encrypted computer code—that when properly "read," allows for a manipulation of reality prescribed by the spell. Mages believe that what they accomplish through the casting of spells does not defy the laws of physics so much as abides by laws not yet known. The execution of a spell relies on the intelligence of the magician, both in his capacity to reason through the complexity of its formulae, and his ability to apprehend its scope in his active memory. Such a task requires tremendous research to understand and decode the spell, and taxing concentration to perform it. In the game, the spellcaster's fortitude for this work is represented by a reserve of magic points (MP) that grow with his experience, an abstraction of the toll spellcasting takes on the spellcaster's mind and body.
To cast a spell, a mage must recite the spell (the semantic component); perform a ritualistic movement (the somatic component), such as a hand gesture, the drawing of geometry on a surface, or a dance; and/or sacrifice a material thing (the sympathetic component): consume a pearl, burn incense, spill blood. Mages stockpile their knowledge of spells in spellbooks and other magical receptacles (such as wands, orbs, or other foci), and as academics, confer with each other on the esoterica of how to catalog and expand upon such knowledge.
Thaumaturgy is the magic of clerics, generally understood to be magic "a Deo,” or magic "from god." Clerics produce magical effects by performing miracles, which are direct interventions on earth by supernatural beings called patrons.
On Patrons & Miracles
A patron may be a spirit, deity, or another extremely powerful being who is at the spellcaster's disposal: whatever the source of the miracle, its effect is supernatural. Through the worship of their patron, the cleric is gifted with a sudden insight—sacred knowledge of what is required to change the world. The collective revelations of a patron's faith, carried by her in scrolls or prayerbooks, holy charms and sacred relics, is a catalog of known miracles, similar to a mage's spellbook.
The effectiveness of a miracle and its relative strength is measured by the cleric's faith, which grows with her experience as she learns how to properly worship the patron. Again, this is abstracted in the game as magic points (MP), as the cleric calls upon her reserve of interventions granted by the patron, until her favor with the being is depleted, and she must renew her obeisance to it (by way of prayer or whatever piety it requires).
Like spells, miracles also have somatic, semantic, and sympathetic components in their casting. What is more important to every cleric, however, is her daily adherence to the tenets of her religion, as her power derives solely from the favor of her patron.
Psionics is magic used by psychics, generally understood to be magic “ab intra,” meaning, magic "from within." Psychics produce supernatural effects by projecting subjective energy from their minds onto their victims. Victims subjected to a psychic's powers have their reality altered because the psychic's subjectivity overpowers their perception of reality.
About Psionic Powers
These powers develop in the psychic as a result of a special connection to an otherworldly spiritual or psychical realm, a connection that can be inherited by birth, created through bonding with an eldritch patron, caused by a near-death experience, or established through a séance that channels the mind's subjective energy as a force for supernatural cause.
More so than any other type of spellcaster, the psychic's body takes a toll for the astral operations of its mind: psychics use their bodies as conduits to transfer subjective energy from the othered realm into reality. Doing so requires the psychic to effectively operate on two planes of existence simultaneously, and their capacity to maintain this duality is abstracted as magic points (MP) in the game, which can only be replenished with deep meditation and rest.
Psychic powers, unlike spells and miracles, have no somatic, semantic, or sympathetic components. The performance of these powers occurs within the othered realm, invisible to the mundane eye (i.e., within the psychic's mind). More complex powers, however, such as seances or summonings, may require sympathetic components, circles of power, or sacrifices, to appease eldritch forces involved in the invocation.