Male Roman citizens would have three names: praenomen, nomen, and cognomen. Generally women would only have one name - the feminine form of their father's nomen - although in earlier times women would sometimes have two names. If there were many daughters all in one family (with the same name) they would have nicknames to distinguish between the siblings, or be named in order (Elder Claudia, the second Claudia, the third Claudia, et cetera)
Praenomen: There are 3 dozen possible praenomina but really only about 17 were commonly used. You can imagine how confusing this could become:
Marcus: Have you seen Gaius today?Or imagine yelling "Quintus!" to your friend in a crowded market and seeing 20 people turn around.
Sextus: Gaius Julius Caesar or Gaius Gracchus?
Marcus: Neither: Gaius Marius.
Praenomina have standard abbreviation; because there are so few, everyone would be well acquainted with what latter represents what name. One matter that confuses Latin students today is that Gaius is abbreviated C and Gnaeus is Cn. Just take my word that the letter G evolved from C and the abbreviations keep the old, traditional spelling.
The eldest Roman son generally would have the same praenomen as his father, but not always. In funerary inscriptions, a Roman's full name is give (Gaius Julius Caesar) and his father's praenomen is given (son of Gaius). It was a given the the rest of the father's name would be the same as his son's. Publius Licinius Crassus, son of Marcus give us enough information to know that his father's name was Marcus Licinius Crassus.
Here are some examples of what a Roman's name would look like on a tombstone or public inscription:
Lucius Hostilius, son of Gaius.
Lucius was probably not the eldest son of Gaius.
For Gaius Oppius, son of Gaius.
Gaius was most likely the eldest son of Gaius, as they have the same praenomen.
Gaius Erucius, son of Gaius
Titus Titius Flaccus, son of Lucius
Titus Titius not only has a super awesome name, but he shares a cognomen (Flaccus) with Quintus Horatius Flaccus, more commonly known as Horace of "Carpe Diem' fame.
Quintus Tullius, son of Quintus