On reaching 0 HP
Once a character reaches 0 HP they are Unconscious, and starts rolling death saving throws on their turn.
(see page 197 of the Players Handbook)
When rolling Death Saving Throws
When rolling there are 4 cases we care about, rolling a 1, rolling from 2 to 9, rolling from 10 to 19, and rolling a 20.
When rolling a 1 you automatically ...
Three death save successes and you're stable but still unconscious
On three successful death saves a PC becomes 'stable'. A companion can also stabilize an unconscious PC with a successful medicine check, or by casting the cantrip spare the dying.
Check the rules for 'Stabilizing a creature' again:
A stable creature doesn't make death saving throws, ...
If you succeed on a Death Saving Throws 3 times, you don't recover any hitpoints. Instead, you become stable:
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
A stable ...
Normally the software rendering you unconscious doesn't want to disconnect you, so that you can still be tracked via your connection. In fact, there are rules for attempts to jack out because of this. Ruling that being rendered unconscious allows one to, effectively, automatically jack out would make being knocked out intentionally a good fallback strategy ...