Just to add a few footnotes to thbz's outstanding writeup:

1: There were a few other (very rare) praenomina - Mamercus, Postumus, and Vopsicus. The praenomina were abbrieviated in Roman inscriptions as follows:

  • A. - Aulus
  • Ap. (App.) - Appius
  • K. - Caeso
  • C. - Gaius
  • Cn. - Gnaeus
  • D. - Decimus
  • L. - Lucius
  • M. - Marcus
  • Mam. - Mamercus
  • M'. - Manius
  • N. - Numerius
  • P. - Publius
  • Q. - Quintus
  • Ser. - Servius
  • S. (Sex.) - Sextus
  • Sp. - Spurius
  • T. - Titus
  • Ti. - Tiberius
  • V. - Vibius

2: Adoption was very common in Ancient Rome, especially for political or pecuniary purposes. When Roman men were adopted they took on the nomen and cognomen of their new father, and frequently added a fourth name (or agnomen) indicating their original gens. Thus when Gaius Octavius (the future emperor Augustus) was adopted by Julius Caesar, he became Gaius Julius Caesar Octavianus, which we shorten in English to Octavian.

3: When Roman men wanted to be very precise, they would give the first names of their father and grandfather and their ancient tribe on top of their own three or four names. Thus the most formal way to identify Marcus Tullius Cicero would be to introduce him as Marcus Tullius Marci filius Marci nepos Cornelia tribu Cicero (Marcus Tullius Cicero, son of Marcus, grandson of Marcus, of the tribe Cornelius).

4: Officially women only had one name: their nomen. Thus, whether Marcus Antonius had one daughter or fifteen, they would all have been called "Antonia." It is true that women were often distinguished by birth order (by adding "Maxima," "Secunda" "Minor," "Tertia," etc.) but with very common family names such as Julia or Cornelia there was still much potential for confusion, as thousands of "Julia Secundas" must have been floating around. One way this confusion began to be resolved in the late Republic was the practice of women taking on the name of their husbands or fathers, in the genitive case, for example Postumia Servi Sulpicii (Postumia, wife of Servius Sulpicius) or Caecilia Metelli ( Caecilia, daughter of Metellus). In the period of the empire, women increasingly began to acquire their own cognomina and even new feminine versions of the praenomina.