Roman names go like this (and bear in mind this is the simplified version). First for blokes: you start with a family name, which comes from your father and is your clan or gens. For example Julius would be a man of the Julii, Antonius would be a man of the Antonii etc. You get this when you're born. If you survive the first 9 days you're given a personal name such as Marcus, but this comes from a very very limited list - 98% of men had one of 17 personal names. Which means that the same bloody names were repeated throughout Roman history and modern researchers end up tearing their hair out trying to distinguish all the different people called Marcus Julius.
To back up the official names you might also get a cognomen, which is sort of a nickname - Caesar means "hairy", Tacitus means "silent", Cicero means "chickpea". As these nicknames eventually became hereditary (after all, sons would be named after their fathers), yet another nickname was added. In fact pretty much all the famous Roman names we know now are actually nicknames.
For women, it was a bit simplier. They didn't have personal names or nicknames, just family ones. So if your father was called Marcus Octavius Africanus you would be called Octavia. So would all your sisters. To distinguish girls they were given numbers: Octavia Prima, Octavia Secunda, Octavia Tertia etc. If anyone outside the family cared, you might also be known as Octavia Africana.