No one wants a geeky answer expanding on that phase, (and inevitably I'm giving it anyway!) but I think it's a good one.
"Republic" is from the latin "res publica" meaning "the things belonging to the people" - in this case, 'the affairs of state'.
In a republic, running the country is something which can involve everyone (most obviously, by voting), it's a public affair. And the opposite arguably is a kingdom, where running the state is a private affair, and ordinary people have less involvement in it, even if eg they're civil servants. (In Latin, it might termed "res privata").
But you can have a country that's run at least in theory by a monarch, such as the UK or Sweden, but which is in practice a republic too. Republic doesn't mean "no king" so much as "everyone can do stuff." Probably can't have a constitutional private kingdom that's run as a democracy though (no matter what places like North Korea might claim!)
