Normally, losing all HP knocks a character out for the rest of the battle. Usually, but not always. If you have enough style points (50 on most difficulties), you can survive with negative HP! This is called Desperation Mode.
In Desperation Mode, damage is increased but HP restoration is reduced. If a character is still in Desperation mode at the end of their turn, they get knocked out. Heal quickly or go out with a bang.