The Roman nickname was known as the cognomen, or the third part of a name, and later the agnomen, or the names following the cognomen. Originally they were nicknames given for a prominent body feature, characteristic, profession, or more notably a military exploit. Eventually the cognomen became part of the standard naming convention as a hereditary modifier of the nomen, or the second name.

The Roman naming system itself was a tripartite system which eventually developed after several centuries. Originally it began with the praenomen and the nomen, a personal and a family name. As the family's grew the use of the cognomen became part of the naming practice and eventually codified into a hereditary title.

Roman custom as to the use of names is similar to know with the praenomen for personal, informal use between friends only. The nomen was for more profesional issues while the cognomen, if an honorific could be used, and if disparaging, left to be used outside of earshot.

Famous examples of the cognomen would be Marcus Tullius 'Cicero', Gaius Julius 'Caesar', Marcus Licinius 'Crassus', and Lucious Cornelius 'Sulla'. These cognomen were family names passed down to each generation, all focusing on a characteristic of its owner or their ancestors.

The eventual formal use of the cognomen created the necessity for another means of distinguishing between famous individuals and families. This gave rise to the use of the agnomen which was used as a nickname just as the cognomen was and some were hereditary however many were because of a particular military action such as Publius Cornelius Scipio 'Africanus'. The agnomen Africanus was granted to him after he defeated Hannibal Barca at the Battle of Zama ending the Second Punic War, a title of immense prestige and honor.

The agnomen could be somewhat more playful as seen with Gaius Julius Caesar Germanicus Caligula. His agnomen was given to him by the armies his father commanded because he wore a soldiers uniform as a child and the Latin diminutive of the soldiers boot is Caligula. his agnomen became extraordinarily popular and he became well known by it.

The imperial names were generally lists of titles and accomplishments such as names of states and peoples conquered. During the second century with the adoptive emperors, the names of emperors gained new meaning as the namesthemselveswere used to provide legitimacy to the emperor. Trajan used this as he included his adoptive fathers name, Nerva' and 'divi' as Nerva had been deified by Trajan granting Him, as his successor, the title. Later the Severan dynasty used a similar practice of posthumous adoptions by previous emperors to gain legitimacy.

Nicknames were immensely important in Roman life as many of the patrician families used the same praenomen throughout generations necessitating the use of a cognomen and agnomen to differentiate the families through the generations.

Benet Salway- "What's in a Name? A Survey of Roman Onomastic Practice from c. 700 B.C. to A.D. 70"

(I would highly suggest giving this a read)

Also I typed this on mobile so please forgive any formatting or autocorrect typos.