Naming conventions changed over time, of course. For the Republican period there was a pretty standard scheme.

Most Romans had three names - praenomen, nomen, and cognomen. Gaius Julius Caesar is an example. Gaius is the praenomen - first name. Julius is the nomen, or clan name, and Caesar is the cognomen, for a subset of the larger clan. Julia, Caesar's daughter, was just Julia, from the clan name.

Some for reasons we don't really understand had only two names - Gaius Marius doesn't seem to have had a cognomen.

Others had more names. There was an agnomen that followed the cognomen in some cases, given because of some particular characteristic or achievement or other notable thing, and this could be passed down. So Publius Cornelius Scipio beat Carthage and became Publius Cornelius Scipio Africanus. His adopted grandson was born Lucius Aemilius Paullus Macedonicus, after his father, but Africanus' son adopted him, changing his name to Publius Cornelius Scipio Aemilianus Africanus. The Aemilianus tells you he's adopted and from where.

Some names were strictly contained within a particular family. The Claudii seem to be the only ones that used the praenomen Appius, for instance. They also had an interesting spelling convention by which, to signal political alignment with... well, for lack of a better term, "the people," they might chose to use the spelling Clodius instead of Claudius, as Publius Clodius Pulcher did.

Freedmen and other people who were made citizens rather than being born citizens tended to take the name of their patron. So we see a lot of Juliuses in the 2nd century CE and going forward. We hear about a Gaius Julius Vindex, governor of Gaul in 68, for instance.

Moving further into the empire I'm on less sure footing, so I'll stop here. I'll also note this is about aristocrats. I'm not sure how far this extends to the lower classes.