The Romans were a Human society on Earth. They had a developed culture and lived in their empire called the Roman Empire from around 500 BC to 500 AD. The Romans and their Empire were centralized in the city of Rome (Roma in their language Latin) located in the future nation-state Italy. Later the capital was moved to Constantinople.
The Romans were eventually defeated by other Humans of another culture which they often called barbarians. The barbarians were the ancestors of the German people, something observed by a Nazi character in a holoprogram set in 1944 France to have been the result of the moral decay of the Romans and the superiority of the Germans. (VOY: "The Killing Game, Part II")
Spock's analysis of the degradation of the cultural development of the civilization found on Sigma Draconis VI, was comparable to the Romans' wars and eventual defeat by the barbarians. (TOS: "Spock's Brain"; VOY: "11:59")
Several scenes of Roman life, including the statue of the founders of Rome Romulus and Remus being suckled by the Capitoline Wolf and a chariot race were seen in the time stream as the timeline realigned itself after Jonathan Archer restored a damaged timeline. (ENT: "Storm Front, Part II")
The proverb "When in Rome... do as the Romans do" often refers to the practice of following suit with what others are doing, particularly by observing and mimicking the customs of citizens of a foreign land the person is visiting. (DS9: "Let He Who Is Without Sin..."; ENT: "Babel One")
In 2268, the crew of the USS Enterprise discovered the planet 892-IV to be inhabited with Romans, mirroring those of Earth. This was evidence of Hodgkin's Law of Parallel Planetary Development. (TOS: "Bread and Circuses")