The phrase "red mage" refers to a character in fantasy games who has abilities in both magic and combat. They usually wear red clothing and can use a mix of offensive spells and weapons. Full definition
As for the new Red Mage class, it's a ranged DPS role with rapiers and magicked crystal mediums as its weapons of choice. (player.one)
Red Mage in particular is a textbook example of how to make a functional melee / caster hybrid in a way I've never seen before. (massivelyop.com)
Everyone who pre-orders Stormblood will receive an adorable Red Mage pet, like the one seen above, and a pair of Ala Mhigan earrings that have +5 of every stat and an increased experience gain for any class under level 50. (thatvideogameblog.com)