Magic classification and theory is a contentious subject, but one of the more common schemes divides magical disciplines into three very broad groupings: Basic Schools, Advanced Schools, and Exotic Schools.
Basic Schools are sometimes claimed to be the most overarching or simplest schools of magic, and are often arrayed on a Great Wheel of Magic, as pictured above. This is far from universal nomenclature, but it's common enough for peasants to be slightly wary of multicolored circles in general, and is often an apprentice mage's first exposure to magic. They tend to be fairly broad and flexible.
Advanced Schools are often regarded as advanced, adjacent, or subservient branches of the Basic Schools, along with the inevitable fury and elitism over which one is best. They vary a fair amount, but are often more focused than the Basic Schools. Notably, each Basic School has a corresponding Advanced School that is typically more combat-focused and blunt than its flexible relative.
Exotic Schools are a grab bag for things that don't fit neatly into the current classification system. This is a trait shared by most if not all magic classification systems, as while many schools suggest obvious relations or qualities, instances of wizards specializing in Explosive Mushrooms or Enchantments Exclusively Relating To Bounciness are distressingly common. They tend somewhat towards the rare or bizarre, but by definition have no overwhelming similarities with all of their fellows.
Basic Schools and their Advanced Schools are listed below. A collection of some of the more common and well-known Exotic Schools follows.
Life Magic: Healing and growth. Can mend wounds, increase vitality and strength, grow something larger, and generally make things more living.
Vitamancy: Healing and vitality. Can heal and bolster living things in powerful but straightforward ways.
Illusion: Mirage and persuasion. Can create illusions and alter perceptions.
Mutationism: Mutation and growth. Can alter living things in new and horrific ways.
Fire Magic: Heat and flame. Can hurl fire, warm things up, intensify flames, and generally make things warmer.
Pyromancy: Destructive power. Can create and direct flame and explosive power, but lacks nuance and fine control.
Consumption: Devouring qualities. Can bolster oneself by eating things associated with the desired effect.
Oathmaking: Binding promises. Can achieve effects by binding things with appropriate restrictions or duties.
Order Magic: Preservation and stability. Can strengthen something's makeup, hold things in place, protect against damage, and generally make things more orderly.
Enchantment: Permanence and connection. Can apply permanent or long-term magical effects to things.
Alchemy: Refinement and expression. Can distill and bring out the magical qualities in things.
Warding: Exclusion and separation. Can prevent things from approaching or affecting locations or objects.
Earth Magic: Earth and stone. Can shape dirt, harden stone, sense things through the soil, and generally make things earthier.
Terramancy: Inescapable power. Can control earth and stone with extreme power, but lacks nuance and fine control.
Arcanomancy: Crystalline power. Can generate and control violet energies sometimes referred to as arcane.
Gating: Bargain and exchange. Can exchange material goods for otherworldly equivalents.
Death Magic: Death and decay. Can rot flesh, animate the dead, sap strength and vitality, and generally make things more dead.
Necromancy: Undeath. Can create and control undead.
Umbramancy: Shadowstuff. Can create and control a dark substance commonly referred to as shadow.
Sacrifice: Bargain and sacrifice. Can create effects by ritually destroying appropriate materials or creatures.
Water Magic: Water and moisture. Can calm waves, generate mist, make water more buoyant, and generally make things wetter.
Hydromancy: Unstoppable power. Can control water with extreme force, but lacks nuance and fine control.
Cryomancy: Gripping power. Can generate and control ice and cold with extreme power.
Summoning: Foreign entities. Can bring forth things from beyond the veil.
Chaos Magic: Corruption and change. Can weaken and subvert a thing's qualities or form, and generally make things more chaotic.
Hexing: Misfortune and failure. Can apply curses and maledictions to things.
Polymorphing: Changing form. Can change self or others into other creatures or objects.
Prophecy: Foresight and prediction. Can gather cryptic information on virtually anything... in theory.
Air Magic: Air and wind. Can generate wind, control weather, lift the caster, and generally make things windier.
Aeromancy: Untouchable power. Can control winds with extreme force, but lacks nuance and fine control.
Sonomancy: Song and sound. Can generate and control sound and sonic vibrations.
Lumomancy: Brilliance and light. Can generate and control light and very bright damaging energy.
Mycomancy: Blunt ability to generate and control mushrooms and fungus. Fungal tissue tends to be sturdy and strong but ponderous and inflexible.
Lightning Magic: Ability to generate and control lightning. Reasonably common for its offensive ability. Sometimes argued to be a subset of Fire Magic or Air Magic.
Heart Magic: Ability to generate and control love and devotion in others, as well as pink energy and crystal. There's some debate as to whether this energy and material is manifested love or what.
Hag Magic: Effects gained through and focusing on inflicting pain and terror. Poorly understood, but tends to empower and twist the caster and grant fell knowledge. Suspected of being vile priestly magic or closely related.
Dagon Magic: Unspeakable fish magic. Poorly understood, but good for warping things into aquatic horrors, gazing upon knowledge not meant for mortal minds, and generally bringing the sea to others. Appears to be a sort of priest magic, possibly aimed at something you're not supposed to derive or invoke priest magic from.
Archarus Magic: Control over hellfire and brimstone, plus the ability to gain or grant demonic boons at a terrible cost. Priestly magic derived from Lord Archarus, one of the better known and commonly worshiped demon lords among the populace. At least, you assume the populace commonly worships demon lords. Who else would they forge pacts with?
Wolf Magic: A totem school focused on emulating the features of wolves. Allows transforming into a wolf, speaking with wolves, inspiring pack ferocity, and other wolfy things.
Spider Magic: A totem school focused on emulating the features of spiders. Allows spinning webs, poisoning foes, turning into a swarm of spiders, and other spidery things.
Sea Cucumber Magic: A totem school focused on projectile vomiting your own innards, gently floating in water, and other sea cucumbery things. Common wisdom is divided as to whether this school's popularity is due to some hidden power only a few can unlock, or because mages really and genuinely cannot resist doing the dumbest thing possible at every opportunity.