We are and always have been a republic. Nowhere in our Constitution, or any state constitution, will you find the word "democracy." In fact, Section 4 of Article IV of the Constitution reads: "The United States shall guarantee to every State in this Union a Republican Form of Government, and shall protect each of them against Invasion...." I would interpret those last seven words to include illegal immigration.
I do not know why or when "democracy" began being used to describe America's form of government. If anyone else does, please educate me. I don't think this is merely a matter of semantics, because a democracy and a republic are two very different forms of government. Democracy means every citizen has a voice/vote on EVERYTHING, much like the government of ancient Greece. Republic means we elect representatives to vote and express our views in Congress/government.