*The examples given here will deal with
Marcus Tullius Cicero and his daughter Tullia
Names For Boys
The name of every Roman man consisted of three parts at least:
The first name ('Marcus' in our example) is the personal name, usually a Roman firstborn will be named after his grandfather. The Romans had a narrow supply of first names (about 30). This name was considered to be the least important name of the three.
The third name ('Cicero' in our example) is the family name or Cognomen, and indicates the nucleus family. It often describes one aspect or another of the family (Cicero for instance means hummus, and it is possible that in some point or another the family made its living from selling hummus). The Cognomen is more important than the first name, as it indicate some of the familial ties of a person, but it is not as important as the second name.
The second name ('Tullius' in our example) is the Nomen Gentis, and is the most important of these names as it indicates the Gens to which the man belongs and thus has great political and social meaning. The Gens (which might be translated as 'clan') is a group of families, connected to each other with historic and mythic ties. The Gens usually goes back to a mythical ancestor and founder (in our case: Tullus)
Any further name of a man is either a title given to him by the state, or (rarely) inherited from his parents. These could be descriptions of political offices that man has carried (for example Gaius Iulius Caesar Dictator), nations he has vanquished (Tiberius Claudius Drusus Germanicus), or significant points in his public service (Quintus Fabius Maximus Cunctator - the hesitant).
In cases of adoption the nomen gentis and the cognomen of a person will change to that of his adopting family, however, to mark the fact that he was not actually born to this gens, his old nomen gentis will be added to his name after his new cognomen incorporating the component -an- (Thus Gaius Octavius Drusus changed to Gaius Iulius Caesar Octavianus)
Names For Girls
Women had no personal name or Cognomen (In the later Republic some married women also began to be referred by a cognomen- usually that of their husbands). They carried only their Nomen Gentis. Thus Marcus Tullius Cicero's daughter was called Tullia. Had he had more than one daughter, his eldest would be called Tullia Prima (the first Tullia) and then Tullia Secunda, Tullia Tertia (second, third) etc.