Instant death happens when there is damage remaining that equals or exceeds your max HP, e.g. if you've got 10 HP, and you take 20 points of damage, you're destroyed.
Aside from the above rule, if you get to 0HP you are unconscious.
A stable creature that isn't healed gains 1HP after 1d4 hours.
Others can use their action to try and stabilise the character with a DC 10 Medicine Check.
Monsters usually just die, they don't roll saves.
Temporary hit points received can't save you.