Health is an attribute of every unit, siege weapon, and structure that indicates how much damage the card can receive before it is destroyed.
Not all damage dealt during combat will affect a card's health. During the cleanup phase, the armor check should be done to determine how much health each card loses.
Once a card loses health, there is no way for it to heal. The only exception to this rule is regenerating lizards, which may remove one damage counter during each Lizard prep phase.