Objectively, an archmage is mage using the Crystal Sphere Techniques to cast a spell that uses two or more spheres at the same time. Later, it would be more precise defined as a mage capable of combining two spheres that were at least ten spherical units apart in a single spell.
However, it is also a popular term with those who are familiar with the different types of magic when describing mages (and occasionally talents) that appear to have a broad range of magical talents.