Damage is the means by which both player and enemy reduce each other's health. When the enemy's health reaches 0, you win and can continue through the dungeon. If the player's health reaches 0, you are defeated and have to start back at the title screen.
There are a few main types of damage. These types are important because some enemies are strong against some types of damage, meaning they take half (rounded up) damage from attacks of specific types. Additionally, some enemies are weak to certain damage types, meaning they will take double damage.
The innate descriptions of enemies are shown before the fight begins to let you know what enemies are weak to elemental attacks and which enemies are strong against elemental attacks.