When talking about political systems you characterize a system using multiple labels.

One label would be to decide whether a country is a monarchy or a republic.

Another would be if it is a autocracy or democracy.

Next you got to talk about whether it has direct rule or representational. If a country is a monarchy, there is a difference between a constitutional monarchy and an absolute monarchy.

The word republic is used today to state that a country does not have a monarch. That is all it means.

Democracy means the people rule. To clarify it further one could say we have representational democracies. We elect representatives as opposed to direct democracy like Athenians. Yet no country today has direct democracy and hence that is a superfelous label.

You can look this stuff on wikipedia or any other encyclopedia or work on political science. The primary reason many Americans are confused about this is that they have elementary school teachers who have been hung up on the US not being a democracy, because that is what the American founding fathers were concerned about. But that is because they have only read American history and not political science as such.

The field of political science does not define republic in this old fashion way. This confusion is a peculiar Ameican centric problem, in large part caused by the US educational system spending so little time on countries outside the US and their political systems.