Romans were the population of Ancient Rome, and New Rome, founded by Rhea Silvia's children Romulus and Remus. They were the enemies of the Greeks, because they conquered their home land, and adapted their gods.
The Roman Empire (Imperium Romanum) lasted for several centuries long, and was one of the most dominant empires of the entire world. It was founded by Romulus and Remus Silvia sons of the goddess Rhea Silvia and Mars. The Romans were supposed to have been the descendants of generations of intermingling between the original Latin tribes of Italy and Trojans who had survived the sack of Troy. Rome's first king was alleged to have been the controversial Romulus, for which the city was named for. During the reign of the seven kings of Rome, the fledgling nation struggled to maintain a regional identity and often warred with its neighbours. During this time, the nobility of Rome, the patricians, were of Etruscan background. The Etruscans were a confederation of city-states in the north of Italy. To the south lay the Latins including the Umbrians and Samnites.
The Roman Republic was founded by Lucius Junius Brutus, who later went on to be one of the two first Consuls of the fledgling Republic.
The Romans conquered everything around them, including the Greeks, making themselves their enemies. Some of their famous leaders where people like Gaius Julius Caesar Octavianus later known as Augustus, Gaius Julius Caesar, Marcus Antonius, Trajan, etc. Rome had a long endurance, and the culture of Rome had a lasting influence on our modern society today. Such as religion, artichtecture, philosophy, law and forms of government. Mainly Europe in particular, as Rome had settled there, governed, particularly Europe, and by means of European expansionism throughout the modern world.
In the Lost Hero, we find out that Jason Grace is a Roman demigod, after a short while in the book. Jason shows up wearing a purple T-Shirt and a strange tattoo, the symbol of Camp Jupiter, and later keeps remembering things about his past. He fights like a Roman, talks in Latin and calls the Greek gods by their Roman aspects. In the end of the book, Juno heals his memory (not fully) and he starts to remember that Camp Jupiter is where he belongs, as he is a Roman demigod.
Percy Jackson is sent to the Roman camp, Camp Jupiter in a trade between Camp Half-Blood and Camp Jupiter's leaders. Percy is Greek, and quickly finds out to survive in the Roman world, he has to act Roman. The Romans are harsh, and not very sympathetic towards him, but he manages to make a few friends such as Frank Zhang and Hazel Levesque The Romans, also fear him, because his father is Neptune.
The Greeks find out where Percy is and send the ship the Argo II to retrieve him. The Romans, are doubtful that they can get along, as Greeks and Romans never had. Reyna the praetor, tells them that they would hear them out and decide what to do. Later, Leo Valdez being possessed by an eliodon starts firing ballistae at the Romans and their city, thus making war between them once again.
Annabeth Chase, later in the book, decides to go after The Mark of Athena and try to find the source of the Greek's anger toward Romans. She believes this can heal the wound between them.
The term Roman is mentioned throughout the book series multiple times, mostly referring to Camp Jupiter and their demigods.