The modern Roman naming system is based on the ancient system called the TRIA NOMINA, meaning "three names". While Roman naming systems varied through time, the modern system is an amalgamation of these ancient variations.
The TRIA NOMINA contains:
The NOMEN. This is the name of the person's GENS; meaning family, tribe, or clan. This is a main name of the TRIA NOMINA
A GENS is a broad familial lineage. Many of these names are believed to come from a specific ancestor in Ancient Rome.
Descendants of these ancestors took on the NOMEN as a way of identifying them as belonging to that person's family.
However, As these families grew and intermingled, along with the fall of ancient Roman society, the exact successors to these family groups became lost.
Today the Senate and People of Rome is working to re-establish these ancient family lineages. If you become a Roman citizen you will be brought into one of these families. This can be done through adoption by current members of a GENS. Alternatively, in the case where there are no existing members, you may be granted permission from the Senate to claim a GENS and become a modern founder.
The GENS you join will determine what your NOMEN is.
The COGNOMEN. This name, which comes after the NOMEN, originally identified unique people in a GENS. Later this name came to indicate a specific branch of GENS. These branches often came from specific ancestors, or a COGNOMEN that had frequent use.
Modern family branches will be organized according to those attested in ancient sources. However, special permission may be granted by the Senate to organize a new family branch in an existing GENS.
Whatever family branch you enter into will be your COGNOMEN.
The PRAENOMEN. This is a name that comes before the NOMEN. Most GENTES have traditional PRAENOMEN that are repeatedly used. Generally, a PRAENOMEN is given to honor a specific relative. For example, a first born son will usually be given his father's PRAENOMEN. Later sons may be given the name of honored uncles. The same can apply daughters and mothers, etc.
The AGNOMEN
As an addition to the TRIA NOMINA there may be times when a fourth type of name is used. This is the AGNOMEN. An AGNOMEN can consist of additional names given for various reasons including special honors, adoption, and additional specificity.
The Senate may give an AGNOMEN to a to honor special deeds of a person.
Even with the TRIA NOMINA there may be multiple people in a GENS with the same name. In these cases an AGNOMEN may be used to specify an individual. For example if two brothers are both named after their father the elder brother may receive the AGNOMEN "MAIOR" and the younger brother "MINOR". Another example could if there are more than two daughters with the same name. They would receive an AGNOMEN in the order of their birth: PRIMA, SECVNDA, TERTIA, etc. Males would also have the same AGNOMEN system, though they would use the male forms: PRIMUS, SECVNVDVS, TERTIVS, etc.
A married woman has the right to utilize her husband's name as an AGNOMEN.
In addition to honorific names, an AGNOMEN could be used to indicate heritage through adoption. In the case of adoption the person will take the TRIA NOMINA of the person adopting them. but they will add an AGNOMEN consisting of their former NOMEN.
Lastly, the AGNOMEN can also be an officially given nickname.