"The United States shall guarantee to every state in this union, a republican form of government ... "
- Article 4, Section 4 of the United States Constitution
Is there a difference between a "democracy" and a "republic"?
The meaning of words tends to change over time, and the words "democracy" and "republic" are no different in this respect. For example, if you were to ask Americans on the streets today whether there's a difference between the two words, many would reply that they are the same (or, at least, close to the same), and some dictionaries even define them as synonymous today. Among them is the website of Princeton University, which offers multiple definitions for each word. One of these definitions is even the same for both words, and their website lists the two words as accepted synonyms for each other in this context. Their shared definition, in case you're wondering, is that they are "a political system in which the supreme power lies in a body of citizens who can elect people to represent them" (Source: entry on "democracy" and entry on "republic") The contemporary use of these words is thus somewhat interchangeable, and the common meaning of these words would admit few (if any) distinctions between them.
James Madison
The Founding Fathers thought there actually was ...
Yet there is a historical distinction between the two that our Founding Fathers recognized. One of them even offered these definitions explicitly in the Federalist Papers. These definitions show a distinction between the two in the mind of this particular Founding Father. This distinction existed in his mind long before the Constitutional Convention, even if he had not yet included it in the Federalist Papers. The Founding Father was James Madison, and he essentially said that a democracy was a direct democracy - or in other words, where people vote on everything directly in person. His phrase for a direct democracy was a "pure democracy," and he defined it as "a society consisting of a small number of citizens, who assemble and administer the government in person" (Source: Federalist No. 10). A good example of this system might be Ancient Athens, the most successful city-state within the lost world of Ancient Greece. Madison's definition of a republic, by contrast, was that it was "a government in which the scheme of representation takes place" (Source: Federalist No. 10). In other words, he said, the definition of a republic is where the people elect others to make those decisions for them. Which is better, you might ask? Are there greater dangers in delegating these powers to our elected representatives, or do the greatest dangers come from other sources, like the "tyranny of the majority"?
James Madison