When you start your turn at 0 hit points, you need to make a death saving throw. Rolling 9 and below counts as a failure, and 10 and above counts as a success. If you accrue 3 failures before 3 successes, you die. If you accrue 3 successes, you are stabilized. If you regain any hit points, both your successes and failures are reset.
When you make a death saving throw and roll a 1 on the d20, it counts as two failures. If you roll a 20 on the d20, you regain 1 hit point.
If you take any damage while you have 0 hit points, you suffer a death saving throw failure. If the damage is from a critical hit, you suffer two failures instead. If the damage equals or exceeds your hit point maximum, you suffer instant death.
Massive damage can kill you instantly. When damage reduces you to 0 hit points and there is damage remaining, you die if the remaining damage equals or exceeds your hit point maximum. For example, a cleric with a maximum of 12 hit points currently has 6 hit points. If she takes 18 damage from an attack, she is reduced to 0 hit points, but 12 damage remains. Because the remaining damage equals her hit point maximum, the cleric dies.
If damage reduces you to 0 hit points and fails to kill you, you fall unconscious. This unconsciousness ends if you regain any hit points.
The best way to save a creature with 0 hit points is to heal it. If healing is unavailable, the creature can at least be stabilized so that it isn't killed by a failed death saving throw. You can use your action to administer first aid to an unconscious creature and attempt to stabilize it, which requires a successful DC 10 Medicine check. A stable creature doesn't make death saving throws, even though it has 0 hit points, but it does remain unconscious. The creature stops being stable, and must start making death saving throws again, if it takes any damage. A stable creature that isn't healed regains 1 hit point after 1d4 hours.