Your total health is a fixed value. It doesn't increase over time and won't regenerate on its own. It depletes when you're wounded in combat, fall from a height, are poisoned or eat spoiled food, as well as when you're exhausted or starving. To heal yourself you have to take a healing potion or sleep. If you're bleeding, you first have to bandage the wound before it will heal.
Your horse's health works very much like your own. However, it can never die – even if its health drops to zero, it will be regenerate eventually.