A mage (plural mages), or magus in archaic form (plural magi) is a person who is knowledgeable in the magical arts. They can be of any race, but in human usage the term commonly refers to humans. Mage is a blanket term for magic users of any power or field, though the ability to cast spells is deemed a defining requirement. In professional circles, "Mage" is an official title for a person who has passed Mage Apprenticeship.
It should be noted that by some effect of using magic, mages tend to live much longer than they would otherwise. Some mages have been able to cheat death for twice the normal lifespan.
Further terms are used to refer to subsets of the blanket term Mage:
Traditionally, a mage's apprenticeship lasts five years. Two of these years are spent helping as lab hands (and, in the past, as lab rats, though almost all magic schools have banned this practice), doing minor tasks in alchemical and magical experiments, as well as studying theory. The third year is spent in silence, which is claimed to help the apprentices focus on their own minds by shutting off outside influence. It is only in the fourth and fifth that the first spells are taught.
Mage apprentices are not technically mages, but once they are in their last two years and can cast several spells, they are often called that.
Apprentice is an official description for a student enrolled in an academy or apprenticed to a practicing Wizard (though the latter is becoming rare).
Once their apprenticeship is ended, students become Independent Mages, and have several options available to them. They may continue their learning and further advance up the ladder, they may become researchers or assistants to Wizards or Archmages, or they may go into business for themselves.
Notably, under the emperors since Stewart I, the restrictive magic policy has forced most "Independent" mages to either remain in their academy or enlist in the Imperial Army. Self-employed mages were practically non-existent under the Hawthorne dynasty, although Empress Prazac I has lifted these restrictions.
Magic includes several fields that have no other purpose than to inflict harm on an opponent or to temporarily improve a person's physical abilities (banned in most sporting contests, but not in duelling). Battle mages have specialized on these fields, which mostly work with elemental powers (and especially fire).
Battle Mage is a professional description, not a title. Most Battle Mages are currently employed by the Empire.
A Wizard is a mage qualified to teach. This is by no means a clear definition. Although magic schools use standardized examinations both for apprentices and for mages wishing to teach, these examinations are so easy that most wizards are vastly overqualified. This problem has become proverbial, such that "failing the Wizardry exam" is an idiom among mages for a blunder of abysmal stupidity. In spite of this, most wizards who do teach are qualified. Those who are not are not hired by any school, and teaching apprentices oneself is becoming an increasingly unattractive business.
Very few mages advance to this level. The "arch" is derived from the three "arcs", or tiers, of mastery. An mage becomes an archmage once he or she has passed the examination for the third arc and produced a masterwork (often a newly invented spell). It is notable that there is not necessarily any requirement on power to become an archmage - knowledge is valued more than power.
After passing this test (or even before then), the archmage will usually withdraw into hermitage for further obscure research.