The basic Roman system of naming was called the tria nomina, or "three names." Take the name Quintus Caecilius Metellus as an example. The first name, the praenomen, was extremely common to the point of being not very useful. There were only about 20 in common use. The second name, the nomen, denoted the family (gens), which could be vast. Our Quintus was a member of the gens Caecilia, an ancient and powerful family. The third name (*cognomen) had traditionally been just a nickname, but had come to serve as a marker of a person's specific branch in large families. So for instance, the vast family of the gens Cornelia had many branches. Consider the names Lucius Cornelius Sulla vs Publius Cornelius Scipio. They belonged to very different "branches" of the same vast family tree.

In the Republic, men had often received (or given themselves) a 4th name (an agnomen) as an honorary title or special distinction. So our Quintus, when he was victorious in battle in Greece, got the honorary "Macedonicus" after his name. So he became Quintus Caecilius Metellus Macedonicus. His relative received a similar title for his victories in Africa, and became Quintus Caecilius Metellus Numidicus. As you can see, many Romans had the same name, and they had clever ways of sorting it all out. The famous Scipio got the honorary title "Africanus" after his victory at Zama against Carthage, and became Publius Cornelius Scipio Africanus.

For the emperors, taking these honorary titles became a matter of pride and frippery. So Caligula took (gave himself? was born with? I'm not sure) the honorary title "Germanicus." Technically, the name implies that Caligula was a man of great accomplishments in Germania. Some of the agnomina tended to stick and be passed down through successive emperors. "Germanicus" is a great example, which all the Julio-Claudian emperors possessed. It became an integrated element of their official title. You can see the list and get a sense of which emperors liked this practice and which did not. Diocletian, for instance, really loved to give himself honorary titles.