Many people confuse 'democracy' with 'pure democracy' (also called a 'true' or 'Athenian' democracy), in which every issue is voted on by every person, every time. Obviously, that's not practical for a nation of 330 million people, so we have a 'representative democracy,' in which we choose representatives from our communities to go, well, represent.
A 'republic' literally means 'public thing.' It means that the government is owned by the people, and in today's context it means that different people are assigned to be in charge of different aspects of the government. The opposite would be a monarchy, where everything is done according to the will of the monarch, or a dictatorship, where the dictator's word is final on everything. Functionally, that means a republic basically is a representative democracy; it would be tough to design one without also making it the other.
We are both a federal republic and a representative democracy, and always have been, thanks to the genius of the Constitution.