The UK is what is called an aristocratic republic* as it retains the Roman Republic's setup: - Sovereign / House of Lords (Senate) / Lower house.

The US Founding Fathers copied England, and Rome's model but replaced the sovereign with a President.

Representative democracies are necessarily republics - although some, like my country, skip the upper house. Rome introduced consuls / republican model because the population was too large, unlike the Greek polis, to be politically manageable. Rome of course was not a democracy.

*or monarchic. I am away from my books so cannot confirm exactly the term. Sorry.