When you are in combat, to the left of your cards it will say "Cards # of #". What does that mean? I couldn't find a reference to it in the player guide.
When a player runs out of cards, they are defeated.
To be technically correct, I would say most likely they are defeated. The player may have cast a damage over time such as Fire Elf or Poison that is still in effect that might finish the monster off, or other players in the same battle might defeat the monster for the one out of cards.