Magician

A magician (often shortened to 'mage') is a practitioner of magic, which is the process of channeling, manipulating, and/or utilizing mana. "Magician" is an umbrella term for many different types of magic users. Spellcasters, alchemists, and enchanters are some examples of magician sub-types.
Spellcasting
A spellcaster, as the name implies, is someone who casts spells; "casting spells" being the process of channeling mana in order to shape it into various practical forms (such as fireballs, physical illusions, creating matter, etc.) or charm objects and creatures (such as making something float, altering someone's perspective, etc.).
There are two types of spellcasters: wizards and sorcerers. Wizards are people born with zero magical talent, who dedicate their time to educating themselves on the techniques of spellcasting. A sorcerer is someone to whom spellcasting comes instinctively, as if it were a facet of their natural physiology (sorcerers are typically elves or neichai).
Alchemy
Alchemy is an off-shoot of arcana and chemistry, wherein the goal is to discover how different materials and substances react when mana is channeled through them. The most common alchemical creations are potions, oils, and magic bombs.
Enchanting
Enchanting is the process of imbuing and object with a magical effect via magic runes. The most common type of enchanting is whitesmithing, the creation of magical weapons and armor.