Short answers, pulled from the story.
Polybius was an ancient Greek historian born around 200 BC in Megalopolis, Arcadia. He is important for writing the Histories, the only Hellenistic historical work to survive in substantial form, and for his political analysis of the Roman constitution, which later influenced Montesquieu, John Locke, and the framers of the United States Constitution.
Polybius was one of around 1,000 Achaean nobles interned in Rome in 167 BC following the Roman defeat of the Antigonid dynasty in the Third Macedonian War. His father, Lycortas, had advocated neutrality during Rome's war against Perseus of Macedon, which attracted Roman suspicion. Polybius remained detained in Rome for 17 years before the Achaean hostages were released in 150 BC.
The Histories covers events from 264 BC to 146 BC, with a particular focus on the years 221-146 BC. The work documents Rome's rise to dominance over ancient Carthage, Macedonia, and the Seleucid Empire, and includes eyewitness accounts of the sack of Carthage and Corinth in 146 BC.
The Polybius square is a cipher device Polybius described in Book X of the Histories that encodes letters of the alphabet using a five-by-five grid with numbers along the top and left side. Each letter is identified by two coordinates, allowing messages to be transmitted as pairs of numbers. Polybius specified that it could be used in fire signalling, where torches were raised and lowered to represent each number, enabling the transmission of any alphabetic message over long distances.
Polybius described Rome as having a mixed constitution that balanced monarchical, aristocratic, and popular elements, preventing the cyclical decay of constitutions he called anacyclosis. His analysis of the separation of powers and checks and balances influenced Montesquieu's The Spirit of the Laws and John Locke's Two Treatises of Government, both of which the framers of the United States Constitution drew upon. John Adams specifically considered Polybius one of the most important teachers of constitutional theory.
Anacyclosis is Polybius's theory, outlined in Book VI of the Histories, that constitutions pass through a recurring cycle of degeneration. A monarchy decays into tyranny, which is replaced by aristocracy, which decays into oligarchy, which gives way to democracy, which collapses into mob rule, and so on. Polybius argued that Rome had escaped this cycle, at least temporarily, because its mixed constitution balanced all three elements against each other.