Hit Points are what determine whether a party member collapses or an enemy is defeated after an attack. Each character in a battle has their on Hit Points, and upon reaching zero they are defeated.
For party members, a downed ally must be revived before they can take any more actions. If the battle ends and they are still downed, then will automatically revive with 1 HP.
For monsters, a defeated enemy goes through their defeat animation (frequently just a "pop") and normally cannot be revived. However, it should be noted that many monsters have Lives that are only depleted each time their Hit Points reach zero, and there are some monsters like Exor that revive after a set number of turns.