Morale is the guage a hero has that fills up when they either perform an action or have an action performed against them. In either case, the action must be offensive in nature and not supportive.
Once the hero(es) morale guage is filled up, they then perform their associated Battle-magic(BM). There are instances where some BM will drain a hero's Morale guage.
For the most part, Valor and Spirit heroes will be the only ones that come with BM abilities. As such, make sure you review the hero, prior to recruiting so you know what they can or can not do in a battle situation.