Dungeon World is a fantastical place. Elementals spirits can be conjured from source materials. Prayers for health, might and divine retribution are successful should the gods be willing. Magic is the name to all attributes possessed by people, places and things that are not derived from mundane sources but rather forces beyond. Bardic song, druidic shapeshifting, the bond a ranger forms with its companion, even the superiority (and shining glow) of a fighter's signature weapon are magical. .
Arcane vs Divine Magic
In its simplest definition, arcane is magic that wizards learn; divine is magic that clerics have gifted to them by their deity. While it is not touched upon in classic DW, infernal magic would weilded by the servants of demons and demons themselves. Arcane magic could be considered naturally neutral or chaotic in alignment; divine magic might be lawful or good; infernal magic is evil by default.
Nature and Death Magic
Classical DW doesn't put any explanation into -why- bards can weave spells through preformance or describe Nature as a supernal entity who might have bestowed druids and rangers with Nature magic and created magical creatures for a purpose- but as a GM? You can certainly explore all these kinds of things with your players through the fiction that develops as you play together. Does Death itself have a hand in the existence of necromancers and the Undead in your world? Ask yourself (and your players) why.
White Magic & Black Magic
Healing and Blessings would be white magic- the magic itself might have the alignment of Good. By contrast, dark or black magic would have the alignment of Evil and be malicious and malevolent by nature. Such would be curses and necromancy.
Magic Types are a holdover from D&D and not explored in classic DW except for tags attached to wizard spells. They are as follows:
- divination: spells that reveal information
- enchantment: Spells that transform the target in some day or grant the caster mental/emotional power over another being
- evocation: Spells that manipulate energy or create "something" from nothing
- illusion: Spells that alter sensory perception or create false images, sounds, etc.
- necromancy: Spells that manipulate, create, or destroy life or life force
- summoning: Spell that instantly bring a creature or object to you