Mage is a general term used to refer to any character or class that can work arcane magic, giving it a more neutral context (or at least informal tone) than arcanist. Whilst it is usually considered more synonymous with the Wizard, it can also cover Sorcerer, Warlock and Witch.
In Advanced Dungeons & Dragons, "Mage" was sometimes used as a specific name for a non-specialist Wizard, attempting to define a generalist caster in the same way as the Abjurer, Conjurer, Diviner, Enchanter, Evoker, Illusionist, Necromancer, or Transmuter.
In Dungeons & Dragons 4th Edition, the Mage is an Essentials subclass/variant class of the Wizard, defined as literally a simpler version of the Wizard that lacks its Arcane Implement Mastery feature. It does, however, bring back some of the classic "Specialist Wizard" angles, choosing a specific school of magic and gaining level-based bonus class features based on that school. When it first appeared in "Hereos of the Fallen Lands", its specializations/schools were Evocation, Enchantment and Illusion. "Heroes of the Shadowfell" added the schools of Necromancy and Nethermancy, whilst Dragon Magazine #391 added the school of Pyromancy.
The 4e Mage works by choosing one school of magic to specialize in at first level; it gains a class feature based on that school, and then a second and third feature at levels 5 and 10. At 4th level, it gains the 1st level school feature of one other magic school of its choice, and likewise gains the 4th level school feature for that second school at level 8. At level 21, it gains the feature "Master of Lore" (+3 to checks for the Arcana, Dungeoneering, History, Nature and Religion skills), whilst at level 24, it gains its final class feature in Master of Spells (when you complete a short rest, swap an unexpended wizard spell for another spell of the same or lower level from your spellbook).
Warhammer "Mages" exist, but are probably not worth distinguishing from Warhammer "Sorcerers" or "Wizards".