Blue Mage worked by basically being a hybrid Melee and Caster. It learned mob spells at random by farming enemies and when the move was used you basically had to kill it and cross your fingers. Once learned you can then equip the spell for use and you had a limited amount of spell slots and a total amount of spell points and each spell had points attributed to it.
Blue Mage was the only job not to have any traits learned through level up and instead created traits through different spell combinations equipped so even though a spell may be lack luster at later levels if it contributed towards a trait it was still useful. Blue Mage also had its own versions of spells ranging from heals (both cure bombs and HoT) Nukes of varying elements and buffs like stoneskin haste and blink (a spell mechanic in XI that generated shadows that were consumed upon being hit and therefore allowing it to take the hit rather than you) but it's greatest spells were physical spells which while given MP costs dealt physical damage of various properties (blunt slashing piercing) and had properties like being multi hits or dealing a poison effect or even acting as a AoE cleave attack.
Late in the game's life they were allowed to learn Notorious Monster moves (boss moves) such as a Behemoth's Thunderbolt which was a wide range AoE thunder nuke with a stun effect or a Vampire's Bloodrake (hard hitting 100% damage dealt absorption). They were also given a merit weaponskill along with the other sword classes that was calculated using physical damage equations but dealt Non Elemental Magic Damage.
It was one of my favorite classes in that game and while it would be hard pressed to carry over the spell system it used I hope they figure out a way to bring it back in spirit but at the very least I hope they keep the Middle Eastern motif the class had going on. It gave a job that was about mimicking enemy signature moves some identity of its own especially since earlier portrayals ranged from a cape and a domino mask to Quistis from FFVIII and Quina from FFIX.