CORE is an energy source that is used as a form of 'insurance' when you upgrade your character through the STAR ratings system. Each core costs GLA 1,000, and combining CORE with GEMs will complete the upgrade. Upgrading your character without the CORE will cause the character to go down by 1 on their STAR rating (if the upgrade is unsuccessful). An exception is for STAR rating 1 heroes, they will always need a CORE to upgrade to STAR rating 2. So make sure you have CORE when you upgrade!