Record Materia are character-specific special abilities that are dropped by enemies from that character's record realm. Keep in mind that the random drop only activates when the character's level cap is successfully broken -- a task that is accomplished by obtaining the character's memory crystal.
Character
Memory Crystal Location
Tyro
FFXIII Gapra Whitewood (Classic)
Black Mage
FFIX Village of Dali (Elite)
White Mage
FFIII Hein's Castle (Classic)
Ranger
FFXII Lhusu Mines (Elite)
Bard
FFVIII Missile Base (Elite)
Josef
FFII Coliseum (Elite)
Kain
FFIII Hein's Castle (Elite)
Rydia
FFIX Burmecia (Elite)
Cyan
FFVI Falcon (Elite)
Cloud
FFXIII Gapra Whitewood (Elite)
Wakka
Event Operation Mi'ihen: Mushroom Rock Plateau (Elite)
Tidus
Event Operation Mi'ihen: Mushroom Rock Valley Floor (Elite)
Vivi
Event: Princess of Alexandria: Ice Cavern Icy Path 2 (Elite)
Garnet
Event: Princess of Alexandria: Ice Cavern Icy Path 3 (Elite)
DK Cecil
Event: Light Against the Darkness: 3rd Station (Elite)
Pal Cecil
Event: Light Against the Darkness: 5th Station (Elite)
Sephiroth
Event: The Jenova Project: Shinra Manor (Elite)
Irvine
Event: Balamb Garden: First Floor Corridor (Elite)
Squall
Event: Balamb Garden: Secret Area (Elite)
Fran
Event: The Dreadnought Leviathan: Nalbina Dungeons (Elite)
Balthier
Event: The Dreadnought Leviathan: Terminus (Elite)
Yuna
Event: Marriage of Convenience: Tower of Light Top (Elite)
Quistis
Event: A Planted Seed: Com Tower Passage (Elite)
Selphie
Event: A Planted Seed: Com Tower Top (Elite)
Edgar
Event: A Fateful Coin Toss: Figaro Castle Engine Room
Sabin
Event: A Fateful Coin Toss: Phantom Train Front Cars
Tifa
Event: The Battle Arena: Mage's Path Heavyweight Arena
As mentioned previously, these materia are given as soon as a character's level cap is broken, or are random drops by random enemies in a character's record realm once a level cap is broken. For example, a Record Materia for Garnet will be dropped by any random enemy from any FFIX dungeon.
Character
Record Materia
Tyro
Attunement I: More dmg when exploiting elem weakness.
Attunement II: More dmg when exploiting elem weakness.
Black Mage
Mana Spring I: Chance to regain 1 black magic ability use.