Roman names at the time of the republic had a three-part formation. The first part was the praenomen (literally "fore-name"), which corresponds well to the later notion of a given name, except that there was only a very small, fixed set of possible praenomina. As time went by, the set shrank even more and particular families might use only a few of those. This made the usefulness of the praenomen as a personal identifier extremely small.
The second part of the three-part name is the nomen (name) which corresponds to our modern idea of a fixed, hereditary surname. This name identified not only which immediate family you belonged to, but identified your connections to a larger extended family.
The third part of the name is the cognomen, which originated as a personal nickname of an individual. However, a sub-branch of a larger family might use a fixed nomen-cognomen combination as their hereditary surname. Additionally, an individual might have more than one cognomen serving different functions; he might have a personal cognomen, a name indicating adoption into some family, and the cognomen of his branch of the family. In a formal name, these cognomens were used together, one after another.
A few special considerations apply to women's names. Originally, there seems to have been a parallel set of feminine praenomina, corresponding to the masculine ones and women would presumably bear one of these along with a feminine version of the nomen. However, by the later time of the republic and the early imperial period, women appear to have dropped the praenomen and used only the feminine version of the nomen, as well as an ad hoc cognomen, usually referring to birth-order in some fashion.
However, under the classical naming system, women at this time did not have praenomina (although traces of feminine equivalents of some of the masculine ones can be found) and daughters were all named with the same feminine version of the family nomen. So all the girls born into a family named Julius would be called Julia. The birth-order names (and other names) were used more in the function of bynames, to distinguish one from another. So with two daughters, one might be Julia Major and the other Julia Minor. And if another was born, the second two might end up Julia Secunda and Julia Tertia instead. By the imperial period, this system was being replaced by a freer choice of something more like true given names for girls.
References:
[1] Johnston, Harold Whetstone, The Private Life of the Romans (Chicago: Scott, Foresman and Company, 1903).