Democracy Or Republic?
Here's an interesting discussion in the Washington Post, 2015, reviewing the history and practice of how the U.S. political process is described.
"Indeed, the United States might be labeled a constitutional federal representative democracy. But where one word is used, with all the oversimplification that this necessary entails, “democracy” and “republic” both work. "
There's a long discussion of representative democracy and of how common usage of terms has come about.
The author's bottom line is the U.S. is a democracy, but it's better to read his article yourself.