During the early Roman Republic men had a praenomen (first name) and a nomen (clan name). By 100 BC a cognomen (family name) was also required on official documents, and when applying for citizenship. Some Romans also had an agnomen (nickname).
In everyday use, a Roman would be known by a combination of his praenomen and nomen, or by his cognomen.
During the time of the early Roman Republic there may have been feminine praenomina, but by the time of the later Republic and Empire women were simply known by the feminine form of their father's nomen. To distinguish more than one daughter of the same father, the words major and minor or an ordinal number were appended to the name.
In the later Roman Empire the feminine form of the cognomen was also given to the daughter.
There are many names of Roman origin in use in Europe today. Most were borne by famous saints and martyrs, which ensured their survival into the Christian era. Some examples are Lucius, Marcus (praenomina), Antonius, Claudius, and Julius (nomina).
Note: the ancient Roman alphabet did not have a J or a U. Instead, they used the letters I and V to double for those sounds. Thus a name written in modern times like Julius was actually written IVLIVS by the Romans.