With this optional rule, when a character succeeds on 3 of their Death Saves they come back to consciousness at 1 HP instead of stabilizing at 0 HP and remaining Unconscious, as though they had rolled a natural 20 on their Death Saving Throw under the normal rules.
Additionally, if a game is using the Heroic Stabilization rule, a character that rolls a natural 20 on their Death Saving Throw may instead immediately spend one of their Hit Dice, if they have any remaining, regaining a number of Hit Points equal to the die roll plus their Constitution Modifier.
Creatures that are stabilized by the standard method through a successful DC 10 Wisdom (Medicine) check, remain at 0 hit points and unconscious as normal, following all the rest of the normal rules for stabilizing a creature.