Chain Skills are powerful attacks that can be used when an enemy gets inflicted with status effects. When the status effect condition is fulfilled, a character portrait suggesting that you can use that character's Chain Skill will appear next to the sword icon.
Chain Skills are unlocked for each hero in the Awakening menu by unlocking surrounding nodes, and the Chain Skill nodes are located in the center. Each hero has a different Chain Skill condition and skill effect. Carefully planning your team to meet those conditions will greatly improve your damage output.