Health is the amount of damage you can take in battle. Health can be increased by increasing your Fortitude. [Clarification/requires confirmation: will Questing Health be increased?]
In Battles, both characters always begin with maximum health. The first character whose health drops to zero or below loses. Characters may have health restored automatically mid-battle if they have purchased potions.
By contrast, damage received while Questing is persistent, and will remain until the character heals by paying gold. If the Health in quests drops to 0, you can't do any more quests until you refill. It is widely recomended to refill at the first mission of book one as it can result in huge savings.
Health does not come into play during Raids.