Magic was a supernatural force controlled by Old Speech and to a lesser extent spells spoken in Hardic runes.
There are various practitioners:
Weather workers: Those who are wind bringers and with power to control the weather, summon mage wind and tame the sea and the winds.
Spell wrights, spell smiths, chanters etc.: With magic being so prevalent on Earthsea there are many skills in magic and vocabulary to match.
Witches, warlocks, sorcerers, sorceresses and enchanters etc: These are magic users who know the lesser arts of magic to varying degree and to a lesser degree true naming and knowledge of the old speech.
The Lesser Arts: the arts of Finding, mending, binding and healing (herbal). the magic of illusions and lesser arts of changing can be included here as well as weather working, songs and chants.
The Greater / Higher Arts: Summoning, changing, naming and patterning.
Wizards or Mages: They know the magic of the users above but also the high arts of the naming, summoning and changing. Wizards gain their staff after time in the Imminent Grove where they learn the balance of nature and how to use magic with care and in harmony as much as possible with the forces of the world.
The Nine Masters of Roke
It is worth noting the nine masters as it shows the main skills of magic users and wizards:
The Master Doorkeeper: He watches and guards the door, only allowing those in who share their true name. However should they come as far as to leave Wizards they have to use his name to leave. In order to gain access or leave, wisdom is needed to attain the answer.
The Master Hand: Teaches the art of illusion, sleight of hand and the lesser arts of changing.
The Master Windkey: Teaches the art of weather working.
The Master Chanter: Teaches songs and stories that teach wisdom and the deeds of heroes.
The Master Herbal: Teaches the art of herbs and healing.
The Master Summoner: One of the higher magical arts which teaches the wizard to summon forces including heat and magnetism as well as other beings which of course has its dangers.
The Master Changer: The higher art of changing and transformation.
The Master Namer: Teaches the true names of things over which a mage can summon, change and bind etc.
The Master Patterner: This teacher is the most enigmatic and teaches the equilibrium of the world and therefore the care needed by wizards who could upset the balance of the world.
Other Magic of Earthsea
The Old Powers of the Earth: Some serve the Old Powers of the Earth, evil spirits bound in a single place. Sorcerers and witches that serve it can help extend its powers and their own, but the old powers mean harm and cannot be controlled.
Female Magic: As book series progresses we see how women can learn magic in themselves have access to deep powers that male Mages have oppressed. Ged predicts a women will in time be Archmage of Roke.
Enigmatic magic: Magic is a great power and whilst what gives a wizard power also sets its limits there are powers and magic not taught on Roke Island. Some sorcerers, witches and Wizards might gain access to some of this but magic is potentially unlimited whereas a humans capcity is limited.