You don't need to play any Ivalice games to understand what's going on, though I'm sure it helps.
To begin, the story centers around a Garlean theater troupe. You meet a young girl, Alma, on the streets of Kugane who implores you to help her; being a hero, you do so, which leads to their airship Prima Vista. There you meet her older brother Ramza and are shortly joined by Cid, ultimately being tasked with finding her missing father in the devastated streets of Dalmasca. Thus begins the first raid, which has you fighting against bandits, monsters, and lucavi as you delve under the city to find the lost city of Lesalia, where the man you're looking for (Jenomis cen Lexentale) has been possessed by a lucavi. After beating it he is left alive, but the auracite is taken by a group of Bangaa brigands led by Ba'gamnan.
Once you get him back to the Prima Vista, Jenomis explains that his mission was to uncover the truth of Ivalice, as it's passed into legend in the modern age and is related to his ancestor Orran Durai (who was condemned as a heretic for writing the truth) whose name he wishes to clear. Some technobabble ensues explaining how auracite works, and in the end you're left to your own devices until Ba'gamnan can be tracked down.
Ba'gamnan's compatriots eventually defect after seeing the negative influence the auracite has on him, appearing on the Prima Vista and informing the group he's retreated to the Clockwork City of Goug on the Ridorana Cataract. Upon arrival you're beset by more lucavi and still-active tower guardians (including the robot Construct 8), and upon reaching the city's apex Ba'gamnan falls prey to the auracite's corruption and transforms into Yiazmat, leaving you with little choice but to put him down. While things appear to have been resolved for now, the auracite has a strange effect on Ramza, and you're left to your own devices until Jenomis completes more research.
Upon your return a lot of bad things happen in succession; Alma is possessed by the auracite and disappears, saying something about blood and a covenant or something (memory's fuzzy). The group collectively comes to the consensus it's talking about the Orbonne Monastery in the Golmore Jungle, but due to that being viera territory it's very hard to get there; only upon striking an alliance with the Dalmascan Resistance's Fran are you able to do so, but Ramza is taken in Alma's stead before you venture far into the dungeon. Fighting your way down it's revealed that the whole raid series was a ploy by Ultima, the High Seraph - a primal-like being - to break the seal placed on it by the original Ramza after Saint Ajora betrayed Hydaelyn by summoning Ultima (both were ancient Warriors of Light; the "blood of the covenant" refers to Warriors of Light, not any sort of bloodline). Aided by Ramza and his compatriots' spirits, Ultima is defeated and banished, seemingly for good this time.
With everything learned, Jenomis has more than enough information to clear his ancestors' names and Ramza takes several levels in kindness after meeting his namesake. Ultima is banished for good, removing a major threat to the world.
The story ends on a stinger - led by a spy within the troupe, the IVth Legion (led by one Noah van Gabranth) is seen within the Clockwork City of Goug, admiring Construct 8 in particular.
This is a basic outline. Very basic. Finer points require inquiry.