In the United States a republic is a non-monarchy with ruled by the people, but it also has the more specific meaning of a state that does not practice direct democracy but rather had the government only indirectly controlled by the people. In the rest of the world this is known as representative democracy.
The use of the word republic to describe representative democracy can largely be traced back to the Founding Fathers and to the early creation of the American republic. At the time of the American Revolution democracy was still associated with the negative views the classical scholars had of it. It was a pejorative term used to refer to what would today be called mob rule. The view was rooted in the writings of Aristotle and others who saw pure majoritarian rule as a form of despotism. Kant believed that a true republic was only one that protected minorities. Thus some of the Founders, most prominently John Adams, proposed that the new nation should be a republic rather than a democracy. The Federalist Papers outline the idea that pure democracy is actually quite dangerous, because it allows a majority to infringe upon the rights of a minority. A republic was thus defined as a state in which the will of the people was at some remove from actual governance. However, some other Founding Fathers used the terms republic and democracy interchangeably.
Using the word republic also tied in with the Founding Fathers' interest in republican ideology and a number of republican ideas were integrated into the new constitution. For instance many see the system of checks and balances being based on the republican belief in mixed government. There is a heated debate among academics as to how important republicanism was to the Founding Fathers. The traditional view was that it was of little import when compared to liberalism. In the 1960s and 1970s a revisionist school lead by the likes of Bernard Bailyn began to argue that republicanism was just as or even more important than liberalism in the creation of the United States. This issue is still much disputed and scholars like Kramnick completely reject this view.
The term republic does not appear in the Declaration of Independence, but does appear in Article IV of the Constitution which "guarantees a republican form of government for the states." What exactly the writers of the constitution felt this should mean is uncertain. The Supreme Court in Luther v. Borden declared that the definition of republic was a "political question" in which it would not intervene. In two later cases it did establish a basic definition. In US v. Cruishank the court ruled that the "equal rights of citizens" were inherent to the idea of republic. In re Duncan it ruled that the "right of the people to choose their government" is also part of the definition. It is also generally assumed that the clause prevents any state from being a monarchy.
Over time this Founders' definition declined as pejorative definition of democracy faded. By the time of Andrew Jackson and the new Democratic Party democracy was seen as an unmitigated positive and it has remained so to this day. The limitations on democracy were slowly removed; the President and Senators were made to be directly elected by the population; property qualifications were removed; and referendums and other forms of direct democracy became widely accepted at the state and local level. At present most people thus refer to the United States and its system of government as a democracy. President George W. Bush speaks about spreading democracies, and not republics, around the world.
Today in the United States republicanism most often refers to the doctrines of the United States Republican Party. The word republic is still mainly used as a synonym for representive democracy.