Morale represents the player's will to fight. If Morale is reduced to zero, much like reducing the player's Fortress Health to zero, that player loses the game.
When a Character Card under your control dies, you lose Morale equal to that card's Morale cost. There are certain Abilities, Character Abilities and Locations that can reduce the opponent's or your own morale.
More can be restored by using certain Abilities or Character Abilities.