Bosses defeated in Adventure can be fought again in Boss Battle.
Boss Battle difficulty is divided into Easy/Normal/Hard/Adverse, and each difficulty is divided into different levels. More difficulties and levels are unlocked after completing the preceding ones.
Bosses have different attack patterns based on difficulty and phase. Read up on attack strategies within the Boss Compendium to figure out their patterns.
Receive special boss equipment that can only be obtained in Boss Battle. Try collecting the set equipment of all Bosses.