Is a republic a state or a country?

Answer 1

Countries are states.

__

Could be both.

If you want to nitpick and get into political theory: Some countries are referred to as states. Iceland is sometimes referred to as a Nation-State. So some States are republics.

And some countries (that are not, by definition a State) are republics.

There is a pretty good description of what a nation state is at Wise Geek. They break it down into understandable language if this confuses you still.