The Archmage is a title granted to the practitioner of the mystic or magic arts who has greater skills and power than all others, and commands a greater portion of the ambient magical energies than any other being on a given world.
By definition, there can be only one Archmage per world at a time. The current Archmage of Earth in the Butlerverse is Pieter Heemskerk, better known among metahumans and mystics alive as The Warlock, but it has been bestowed upon other mystics in the past. Warlock has held the position since the abdication of his mentor, The Lady.
While there is no sign that Warlock is considering stepping down, most mystics on Earth consider the Saudi mystic who calls himself Ibrihim to be the Archmage in waiting. The Aspirant, a Romanian mystic of great power, has actually demanded Warlock step down so that he can assume the office himself. Naturally enough, nothing came of this demand.