Overseer of the Damned would trigger 3 times and you would create 3 2/2 zombies when those triggers resolved.
There are a few different ways that triggers are scoped in magic, I'll look at three of them here that all create 2/2 zombie tokens. I put the trigger in italics and what the trigger is watching for in bold. First we have the example here for Overseer of the Damned:
Whenever a nontoken creature an opponent controls dies, create a tapped 2/2 black Zombie creature token.
Second we have Curse of Shallow Graves (There are no death triggers with wording like this):
Whenever a player attacks enchanted player with one or more creatures, that attacking player may create a tapped 2/2 black Zombie creature token.
And third we have Doomed Dissenter:
When Doomed Dissenter dies, create a 2/2 black Zombie creature token.
The difference is what they watch for, how broad or narrow. Overseer of the Damned is the most broad, watching every creature (that fits criteria), so when any of those creatures die, it triggers for each creature, so you create 1 zombie per creature. The second example watches for "one or more" creatures. It doesn't care if it was 1 creature or 100 creatures, it will only trigger once and create 1 zombie. The third references the card by name, so it only watches itself, and will only create 1 zombie.