Magic systems

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Slot-based
Spells are prepared in advance, and once prepared a spell can be cast at anytime with little or no chance of failure, but once cast the spell is consumed; to cast a spell twice it has to be prepared twice, and so on.
Typically the maximum number of spells that can be prepared is finite, either with an explicit cap (i.e. 'Slots'), or by requiring a finite resource (i.e. Material components).
Variations include: more powerful spells require more slots/resources; D&D-style leveled slots; pre-requisite spells to implicitly consume extra slots for powerful spells.
Mana-Based
Spells consume a quantity of spell points (i.e. Mana) when cast, otherwise spellcasting is unrestricted.
Variants are: 'Aspected Mana' (Fire mana, water, defensive, aggressive, etc., and other more unusual divisions); spells consuming things other things instead of mana (hit points, material components, food, etc.), and spells consume more then one type of thing/aspect of mana (e.g. 3 fire mana and a stick of charcoal)
Chance-Based
Spellcasting is essentially free, but has a chance of failure, probably a very large chance.
A variant is to have the chance of failure be affected by some combination stats, skill, or environment.

It is quite common to combine more than one of these methods.

A Slot/Mana system might let you memorize a shortlist of spells, which are then cast as a mana system. Crawl does this. A Mana/Chance system might allow skill to reduce the mana cost of casting spells. The Pen and Paper GURPS does this.

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