A republic is a way to organise a state and its government (with elected head of state and a kind of legislative or governing assembly). In a representative democracy, the people doing the governing and the legislating are elected by the citizens who grant them legitimacy.

You can have democratic (e.g. Iceland) or oligarchic republics (e.g. Ukraine), all the way to dictatorships (e.g. Russia).

Being a democracy and being a republic are completely orthogonal.