An agnomen (Latin: [a?'no:men]; plural: agnomina), in the Roman naming convention, was a nickname, just as the cognomen was initially. However, the cognomina eventually became family names, so agnomina were needed to distinguish between similarly named persons. However, as the agnomen was an additional and optional component in a Roman name, not all Romans had an agnomen (at least not one that is recorded).
propria hominum nomina in quattuor species dividuntur, praenomen nomen cognomen agnomen: praenomen, ut puta Publius, nomen Cornelius, cognomen Scipio, agnomen Africanus.
(Men's personal names are of four types, praenomen, nomen, cognomen and agnomen: praenomen for instance Publius, nomen Cornelius, cognomen Scipio and agnomen Africanus.)
Marius Victorinus further elucidates:
Iam agnomen extrinsecus venit, et venit tribus modis, aut ex animo aut ex corpore aut ex fortuna: ex animo, sicut Superbus et Pius, ex corpore, sicut Crassus et Pulcher, ex fortuna, sicut Africanus et Creticus.
(Now the agnomen comes from outside, and in three styles, from personality or physique or achievements: From personality, such as Superbus ["Haughty"] and Pius [displaying the Roman syndrome of virtues including honesty, reverence to the gods, devotion to family and state, etc.], from physique, such as Crassus ["Fatty"] and Pulcher ["Handsome"], or from achievements, such as Africanus and Creticus [from their victories in Africa and on Crete].
As a minimum, a Roman agnomen is a name attached to an individual's full titulature after birth and formal naming by the family. True Roman nicknames, fully replacing the individual's name in usage, are rare. One such example where the nickname fully replaced the individual's name in usage was the Emperor Caligula, where that name was used in place of, and not along with, his full name, which was Gaius Julius Caesar Augustus Germanicus. Caligula's praenomen was Gaius, his nomen Julius, his cognomen Caesar. Some agnomina were inherited like cognomen were, thus establishing a sub-family. Caligula's agnomen came from the little boots he wore as part of his miniature soldier's uniform while accompanying his father Germanicus on campaigns in northern Germania. In turn, Germanicus received his agnomen in 9 BC, when it was posthumously awarded to his father Nero Claudius Drusus in honour of his Germanic victories. At birth, Germanicus had been known as either Nero Claudius Drusus after his father or Tiberius Claudius Nero after his uncle. As with Caligula, Germanicus is mostly referred to by his agnomen.
An agnomen is not a pseudonym, but a real name; agnomina are additions to, not substitutions for, an individual's full name. Parallel examples of agnomina from later times are epithets like Thomas Jonathan "Stonewall" Jackson (though he is known more often by his agnomen than his first name) or popular nicknames like "Iron" Mike Tyson or Earvin "Magic" Johnson.