Roman Names - Clans
The Romans were divided into various clans, (gentes,) and each clan into several families. Those of the same gens were called gentiles, and those of the same family, agnati. But relations by the father's side were also called agnati, to distinguish them from cognati, relations only on the mother's side.
Roman Names - the Praenomen
The Romans had three names, to mark the different clans and families, and distinguish the individuals of the same family - the praenomen, nomen and cognomen. The praenomen was put first, and marked the individual. It was commonly written with one letter; as A. for Aulus: C. for Caius, sometimes with two; as Ap. for Appius. Not all of the Roman names were used; commonly just two names, and sometimes only the surname. But in speaking to any one, the praenomen was generally used as being peculiar to citizens, for slaves had no praenomen. (See the article on Roman Literature for lots of examples of famous Roman Names)
Roman Names - the Nomen
The nomen was put after the praenomen, to mark the gens, and commonly ended in ius; as Julius, Cassius, Cornelius, Fabius.
Roman Names - the Cognomen
The cognomen was put last, and marked the family; as Cicero, Caesar. The surnames were derived from various circumstances, either from some quality of the mind; as Cato, from catus, wise: or from the habit of the body; as Calvus, Crassus, etc. or from cultivating particular fruits; as Lentulus, Piso, &c. Quintus Cincinnatus was called Serranus, because the ambassadors from the senate found him sowing, when they brought him word that he was made dictator.
Roman Names given to sons and daughters
The praenomen was given to boys on the ninth day, which was called dies lustricus, or the day of purification, when certain religious ceremonies were performed. The eldest son of the family usually received the praenomen of his father. The rest were named from their uncles or other relations. When there was only one daughter in the family, she was called by the name of the gens: thus, Tullia, the daughter of Cicero; and retained the same after marriage. When there were two daughters, one was called major, and the other minor. If there were more than two, they were distinguished by their number; thus—prima, secunda, tertia etc.
Roman Names - The Fourth Name
Sometimes there was also a fourth name, called the agnomen, added from some illustrious action, or remarkable event. Thus, Scipio was called Africanus, from the conquest of Carthage and Africa: for a similar reason, his brother was called Asiaticus.
Roman Names - The Freeman
Those were called liberi, free, who had the power of doing what they pleased. Those who were born of parents who had been always free, were called ingenui. Slaves made free were called liberti, in relation to their masters; and libertini, in relation to free born citizens.