Romans often had three names, the praenomen, nomen, and cognomen. The nomen was a family name, and relatives who shared it were a gens.

Lets use the most famous Roman, "Caesar". His full name was Gaius Julius Caesar, Julius being his nomen, as he was of gens Julia.

Ones gens was very important when it came to social standing. Some gentes were purely plebian, while others (like gens Julia) were high patrician.

Today people know the gens, or family names of most famous Romans even if they don't know what it means.

What throws us off is the cognomen or third name that many Romans had. At first the cognomen was a nickname, sometimes unofficial and sometimes granted officially. For example it is posited that Caesar at one time meant something along the lines of "Hairy". Eventually, cognomen often became hereditary and were sometimes used to differentiate different branches of a gens.

To add a quick answer for the Greek side of things:

In the Classical and Hellenistic periods, there was no "family" name for most Greek communities. The most famous and powerful families sometimes had a name (like the mythical Geomoroi of Corinth, or the Alkmaionidai or Eteoboutadai of Athens), but mostly the Greeks used patronymic and social grouping conventions.

A patronymic works like this: Bob son of Tom, or Alkibiades son of Kleinias. Rarely, the father and the son had the same name (like Demosthenes son of Demosthenes). Often, names would repeat every other generation. So a kid and his grandfather would share a name, and then two generations later the name would appear again.

In some communities, a third "name" was added that identified a religious or social group membership (like a tribal membership, or a phratry). In Athens, the territory of Attica was divided up into smaller administrative units called demes (much like counties in American states). This name was frequently attached to a name to further identify people: Alkibiades son of Kleinias of the deme Skambonidai. This third name is called a demotic.

Sometimes, famous Greeks could come to be called by their city of origin or residence (Dionysus of Halicarnassus, for instance, or Xenophon "the Athenian").