C.cores are special thingies that Gives bonuses/upgrades to your particular Party member whenever they level up
. AD skills are connected to these c.cores. So if you want to earn a lot of AD skills for your character, make'em wear C.cores and make'm level up! the best c.Core is the Tutti c. core obtained from stealing from Nebilim. If equipped, it gives out a whooping +4 bonus points to P.Atk, P.Def, F.Atk, F.Def, Agility and Enhancement. You can check how many bonus points your character obtain from the c.cores in the C.cores menu ^^.
sorry for the loooong explanation but hope this helps you in your game ^^