A classical republic, according to certain modern political theorists, is a state of Classical Antiquity that is considered to have a republican form of government, a state where sovereignty rested with the people rather than a ruler or monarch. These include states like Sparta, Athens, and the Roman Republic. The Romans used the term res publica to describe their state, but the most common sense of that term is closer to body politic or commonwealth. The phrase was coined, it seems, to distinguish the post-Tarquin political system with the previous monarchy, the res privata.
The idea of republicanism was a creation of the Renaissance. The Renaissance scholars, most prominent among them being Niccolò Machiavelli, looked back on the ancient period with great interest and reverence. They defined republic as any state that was not headed by a monarch - thus including the Spartan diarchy, the Athenian democracy, and the Roman res publica. The Italians, themselves living in Republics like Florence and Venice (although note that the doge was an elected monarch), looked back on these states as models of social organization. They looked to the history of the classical republics and attempted to emulate their model. In particular, they saw the mixed government of Rome and Sparta as the secret to stability and the pursuit of civic virtue as the key to the citizens' well being. The theory of government based upon this Renaissance study of the past is known as classical republicanism. Other elements the classical republics shared was the central importance of citizenship. The percent of the population that were citizens was quite limited, but they also had important burdens such as military service. The focus on civil virtue also meant that little attention was paid to individual liberties in these states.
The term classical republic is also sometimes used to describe those later states that are argued to follow the classical model. Thus the United States is sometimes called a classical republic.