The United States is essentially a Democratic Republic. In Ancient Greece, they were a pure Democracy, where the citizens voted directly on every choice in government. So, the Democratic aspect of the U.S. government reflects that of Ancient Greece.
However, it differs because the U.S. system is not a pure democracy. A Democratic Republic is a system where the citizens vote directly (Democratic) for others to represent them in government (Republic). Americans vote for representatives and senators in their respective districts and states, as well as the President of the entire country. Citizens are not directly voting on policy, but the people they voted in do, and in most cases, these representatives will reflect their constituents’ views in government.