Sen. Mike Lee is, of course, technically correct that the United States is a Republic rather than a Democracy. The Constitution itself "guarantee[s] to every State in this Union a Republican Form of Government." (Art. IV, Sec. 4) We solemnly "pledge allegiance to the flag of the United States of America, and to the Republic for which it stands."
Our Founders never used "Democracy" and "Republic" interchangeably-they understood the difference and warned against the extremes of democracy.
In Federalist No. 10, Madison warned that a democracy, in which citizens assemble and administer the government in person, produces "turbulence and contention," while a republic provides a scheme of representation that tempers and refines the emotional and fleeting demands of public clamor and produces results "more consonant to the public good than if pronounced by the people themselves." (See also Federalist No. 14)
Jefferson observed that "democracy is nothing more than mob rule, where 51 percent of the people may take away the rights of the other 49."
So, why do some refer to our form of government as a "democracy"? Possibly because the dictionary defines "democracy" as "government by the people," which accurately describes how "[w]e the people" try to govern ourselves (remember Lincoln's Gettysburg Address).
To reconcile the two terms, we might view "democracy" as a more general description and "republic" as a more specific, refined, or higher form of democracy. We the people govern ourselves through elected representatives. Our Constitution reinforced this representative or "Republican" form of government by providing for the selection of senators by the state legislature and the selection of the President through the Electoral College.
The purpose was to prevent the rise of powerful and persuasive demagogues who could play on prejudices, false claims, and promises to sway the populace and obtain public office. Our Republic was weakened by the 17th Amendment, which altered the selection of senators, and would be further harmed by abolishing the Electoral College, as some demand today.
I do have one significant correction, though, for those who describe America, from the perspective of the federal government, as a mere "Republic." More accurately, we are, from the perspective of the States, a "Compound Republic," as described by Madison in Federalist No. 51: "In the compound republic of America, the power surrendered by the people is first divided between two distinct governments, and then the portion allotted to each subdivided among distinct and separate departments. Hence a double security arises to the rights of the people." Federalism, the division of powers between national and state governments, is thus essential to protection of individual rights.
While we govern ourselves through elected representatives, the people retain ultimate control of the government and laws through the power of their vote in the election process. That is the one right we must zealously protect and never minimize nor neglect. Our informed vote is essential to our Republic.