The term Archmage is used to describe any Mage capable of personally casting at least one arcane spell deemed to be among the most powerful. The mage must personally have the knowledge or ability to cast the spell, as opposed to those with a powerful staff or other magical device capable of casting them.
DM’s Note
As a practical matter, the term Archmage is meant to describe sorcerers, warlocks, and wizards of 17th level or higher, the level at which the mage can cast 9th-level spells.