Roman personal names
From the early years of the third century BCE upper-class male Romans usually had three names, e.g., Gaius Julius Caesar (101-44 BCE), the general, politician, and writer widely known in English as 'Julius Caesar'; Marcus Tullius Cicero (106-43 BCE), the orator and writer ('Cicero'); and Publius Vergilius Maro (70-19 BCE), the epic poet ('Virgil').
The second name - in Latin simply the nomen, 'name' - was adjectival in form and indicated the clan (gens) to which the individual belonged. Thus Gaius Julius Caesar was a member of the Julii, or Julian gens, while Marcus Tullius Cicero was a member of the Tullii, or Tullian gens. A clan was a group of families linked by their shared name and shared belief in descent from a common ancestor.
The third name - in Latin the cognomen (from co-, together, and nomen, name) - was the name of the family (domus) to which the individual belonged.
Very occasionally, an ancient Roman had a fourth name - in Latin the agnomen (from ad-, in addition to, and nomen, name). This was usually a nickname, conferred in virtue of a distinctive personal characteristic or memorable exploit. The most famous example is that of Quintus Fabius Maximus Verrucosus Cunctator (died 203 BCE), who had two agnomina: the first, Verrucosus, (which means 'Warty') is self-explanatory, while the second Cunctator (which means '˜Delayer') was acquired through his dogged insistence on a strategy of delay and refusal to do battle with the Carthaginian army under Hannibal - a strategy which eventually resulted in victory for the Romans. (Fabius' success is celebrated in a well-known hexameter line by the epic poet Ennius (239-169 BCE): Unus homo nobis cunctando restituit rem, One man by delaying rescued the situation for us.)
Ancient Romans are usually referred to in English either by their nomen or by their cognomen. Thus Caesar is referred to by his cognomen, as is Cicero - though in earlier centuries Cicero was more commonly known as Tully, i.e., an anglicised form of his nomen was used; Virgil too is referred to by an anglicised version of his nomen. There is no rule to tell you what is correct in the particular case: you must simply discover, and follow, the convention.
Women and slaves in ancient Rome normally had a single name. This was followed, if necessary, by the name of the adult male (e.g., father, husband, master) to whose authority they were legally subject.