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Killer: Hit Human weakness -> Dmg up ---Richard: Fame and Faith 
Crusher: Hit Inorganic weakness -> Dmg up ---Asbel: Isleberg  ---Sophie: Titan's Knuckles 
Exterminator: Hit Insect weakness -> Dmg up ---Sophie: Garm's Fangs  ---Cheria: Assassin's Daggers  ---Pascal: Genius Staff 
Mechanic: Hit Machine weakness -> Dmg up ---Sophie: Crystal Cutters  ---Malik: The Scrapper  ---Pascal: Staff of Expiration 
Saint: Hit Nova weakness -> Dmg up ---Asbel: Excalibur  ---Sophie: Shining Talons
Gardener: Hit Plant weakness -> Dmg up ---Cheria: Fruit Knives 
Saurian: Hit Reptile weakness -> Dmg up ---Asbel: Ice Coffin  ---Sophie: Lizard Rippers 
Necromancer: Hit Spirit weakness -> Dmg up ---Cheria: Keris Blades  ---Malik: The Undertaker  ---Pascal: Uroboros 
All of them would start Level 1 at +20% and increase by +4% per level until Level * at +56%
Certain dropped weapons can also give these stats at random. That's the only way to get the Eraser effect or more than one Necromancer (the weapons listed come from the named golems in Lastalia Shaft).
Haven't gotten around to figuring out armors or the Accel effects yet, sorry.