Your health is tracked by hearts in the upper right corner of the screen. You start out with three hearts, and through leveling up, you can increase that number to ten. You can lose health from being attacked by animals when outside of town, or by becoming overwhelmed at extreme temperatures in desert or snowy regions. Different animals can do anywhere from half a heart of damage to two hearts of damage. When all your hearts have been depleted, you see the game over screen and start from the entrance you used to that area. Also, all items in your normal inventory will be gone.
In addition to normal attacks, you can also be stunned and poisoned. Being stunned makes you move very slowly until you move around enough for it to wear off, and poison constantly drains your life at a slow pace. You'll know you're stunned because your hearts turn yellow and you'll see big lightning strikes next to your head. When you're poisoned, your hearts turn purple and purple spirals come floating off your head.
To heal yourself, you need to either take a healing item or get some rest in your bed or at an inn. Medicinal Herbs will heal a little bit of health, Potan's Medicine a lot of health, and Anc Water will fully restore your hearts. In order to get rid of poison, you will either need to take Obaba's Antidote or rest in a bed. You can also protect yourself from overheating or freezing by taking Really Cold Ice Cream or Hot Drinks, respectively. To avoid having to use your items up, be sure to drink at the oases in deserts and bathe in hot springs in snow fields.