The argument
A mixed constitution (Greek politeia mikté, Latin res publica mixta) is a regime in which the powers of monarchy, aristocracy and democracy are distributed across distinct institutions and forced to check one another. The classical claim, made most clearly in Polybius VI, is that no pure form is durable: each — when it stands alone — decays into its corrupt double. A regime that holds the three elements in tension can resist that arc much longer than any of them in isolation.
The argument is not original to Polybius. Aristotle already in Politics IV treats the politeia as a mixture of oligarchy and democracy, and Plato's Laws proposes a constitutional mixture in place of the Republic's ideal city. What Polybius adds is the Roman case: a working constitution that, on his analysis, divided supreme authority among the consuls (monarchic), the Senate (aristocratic) and the assemblies (democratic), and which therefore had survived where Greek constitutions had failed.
The Roman inheritance
Cicero in De Re Publica takes the Polybian analysis and makes it Latin and explicit. His Scipio defines the res publica as res populi — the people's affair — and argues that the genuinely stable constitutional form is the mixed one, of which Rome is the example. The argument was already partly elegiac by 54 BCE, when Cicero began the work: the institutions he was describing were straining under the weight of Marius, Sulla, Pompey and Caesar. But the idea of the mixed constitution outlived the polity it described.
The long transmission
Polybius VI and the Ciceronian inheritance are the most consequential classical influence on later European constitutional thought. Machiavelli's Discorsi opens with an extended commentary on Polybius' regime cycle. Montesquieu's De l'Esprit des lois reads the separation of powers through the same frame. The American founders — Adams' Defence of the Constitutions above all — return to Polybius and Cicero explicitly. The architecture of the federal constitution they wrote is a mixed-constitutional one in this classical sense.
Why this still matters
The mixed constitution is one of the few large ideas about how political life can be ordered that survived from antiquity into modernity without being abandoned. Whether the modern institutional form actually preserves the substance the classical theorists meant by it — distinct centres of authority, each willing to use its powers against the others — is a separate question, and one every later constitutional debate has had to answer in its own terms.