Down is the "ultimate" status effect in most RPG games. When a character's HP reaches zero, he or she falls in battle, then lays motionless on the ground. A "downed" party member cannot attack or use any commands. After a battle is won, any downed party members are restored to 1 HP (except Paper Mario: The Thousand-Year Door, where that HP remains zero), excluding bosses, when all party members are restored to full HP after being defeated.
History
Super Mario RPG: Legend of the Seven Stars
Party members are effectively out of the battle unless they are revived. either via a Pick Me Up or Princess Toadstool's Come Back technique. If all three party members are downed in battle, it's Game Over for Mario.
Mario & Luigi series
When a brother's HP reaches zero, he falls in battle, and then is no longer able to battle until he is revived, via a 1-Up Mushroom, a 1-Up Super, or a 1-Up Deluxe.
Paper Mario: The Thousand-Year Door
If a partner's HP reaches zero, he or she falls in battle. They cannot act in battle or be revived for the rest of the battle and must be switched out.
A "downed" partner's HP are not recovered to 1 HP after a battle is won, excluding all bosses except Shadow Queen, when all partners are restored to full HP after falling in battle.