In simplest terms, a Red Mage is a JRPG character who uses both healing magic and damaging magic.
In Japanese roleplaying games such as Final Fantasy and Bravely Default, typically the main player characters are able to select a specific job class which has a narrowly defined set of abilities, and the player is encouraged to manipulate the character's stats, such as their physical strength or their intelligence, in order to make the most of their specialized skill set. This usually means sacrificing all other character build options, in favour of specialization, and versatility must be obtained through the collective mutually-supportive roles of several characters with non-overlapping skill sets.
The Red Mage is a conditional exception to this system, because it combines the features of many classes simultaneously - healing, elemental offensive magic, sometimes even the wearing of light armour and the use of a finesse weapon like the rapier - at the expense of being less than expert at all of these skills. The phrase "jack of all trades; master of none" is applied most accurately to the Red Mage, and often this job class will enjoy exclusive ability bonuses that only activate when more than one type of magic is used in consecutive turns of a battle.
The Red Mage generally becomes available very early in the course of a JRPG, because it serves as a dependable temporary class: one's entire party can be simultaneous Red Mages, in order to face down some opponents who deliberately single-out more specialized targets (such as targeting the healer first to put the party on time constraints to finish the battle. Later in a campaign, however, the Red Mage's lackluster upward limit to any one skill becomes a liability, and the class is abandoned for specialist classes.
Red Mages also tend to have the weaknesses of every class incorporated into their melange of skills, and these weaknesses are not necessarily tempered proportionately in the same manner their skills are capped lower than those of specialists. This means that as the game wears on and boss fights become more and more tactically complex, a Red Mage becomes harder to keep alive on the field, and classes with fewer and more finite weaknesses become more sustainable.
Iron Noder 2017, 29/30