The Twelve Legendary Arms are the endgame gear: 8 weapons, 3 armours and 1 accessory. Each one exists first as an ancient piece that you find somewhere in the world, and each ancient piece is what you offer at the matching altar in the Chamber of Arms on Dirnado to awaken the named arm.

FINAL FANTASY RESONANCE legendary arms: every piece and its altar

ArmTypeWho can equip itGuardianWhere the ancient piece is
RagnarokGreatswordRainAlphergFirst win of Colosseum Fledgling Battle 3
Sasuke's BladeKatanaLasswellAldhaferaIn the Crystal Tower
Artemis BowBowFinaDabihGiven at the Chamber of Arms itself
Thor's HammerHammerLidTegmineHidden treasure No. 6 of the Mysidia Radar, on Dirnado
Last ResortTomeNicholElnathSold by a merchant ship for 45,000 gil
ArcturusFirearmJakeAlhenaOblivion Palace chest at 76 Tablet Shards
Wizard's RodRodSakuraSheratanIn the Abandoned Cabin in Pharm
Fire BirchWhipDark FinaNunkiOblivion Palace chest at 84 Tablet Shards
Tatenashi ArmorCuirassRain, Lasswell, Lid, JakeSadalsuudFirst win of Colosseum Brawler Battle 5
Mirage VestGarmentAny of the eight heroesVindemiatrixHidden treasure No. 2 of the Mysidia Radar, on Gronoa
Lordly RobesCloakFina, Nichol, SakuraShaulaSold by a merchant ship for 40,000 gil
Gold HairpinAccessoryAny of the eight heroesBrachiumIn the Quarry Cave in Zoldaad

The Mysidia Radar needs the magi-tenna picked up in Gaberada in chapter 6, after which you talk to Lid aboard the airship and the radar's hidden treasures appear on the map.

Offering the piece at the altar

Each piece needs its trial at level 50 to awaken and its ultimate trial at level 75 for the "+" form. On the eight weapons, that "+" form raises the damage cap to 99,999, which is what makes the endgame fights possible at all — without it, every hit stops at the cap long before the HP bar moves. The trial rules and every guardian's weakness are on the chamber of arms page.

Some pieces come from shards

Two of the twelve are paid for with Tablet Shards: the Ancient Firearm at 76 shards and the Ancient Whip at 84 shards, both in the Oblivion Palace. Those two pieces also count toward the Legendary Arms trophy, and the two Adamantite from the Oblivion Palace count toward The Weaponsmith, so 89 Tablet Shards cover every trophy in the game. Where all 99 shards hide is on the tablet shard page.