Short answers, pulled from the story.
The tria nomina is the combination of praenomen, nomen, and cognomen that Roman grammarians treated as the defining mark of Roman citizenship. Although all three elements existed throughout most of Roman history, not every citizen used all three names at the same time.
The binomial pattern of praenomen and nomen was already established in Latium and Etruria by around 650 BC. The fuller system later associated with Rome developed continuously from at least the seventh century BC to the end of the seventh century AD.
The praenomen was a personal name that distinguished individual family members, while the nomen was a hereditary surname marking someone as part of a specific gens. Sons typically shared their father's nomen but received different praenomina.
A cognomen ex virtute was a surname earned from a heroic or notable deed, such as Marcus Valerius Corvus, who took his name after defeating a giant Gaul in single combat aided by a raven. Other examples include Gaius Marcius Coriolanus, named for capturing Corioli, and Publius Cornelius Scipio Africanus, named for defeating Hannibal in the Second Punic War.
In AD 212, the emperor Caracalla's Constitutio Antoniniana granted citizenship to all free men in the empire, and nearly all of the newly enfranchised took the nomen Aurelius. Aurelius became the most common nomen in the eastern empire and the second most common, after Julius, in the west.
The system broke down as Roman institutions collapsed through the sixth century, and by the end of the seventh century AD the people of Italy and western Europe had reverted to single names. Some names from the old system survived into later European naming practices.