As in their names in contemporary times. Looking at the "List of Roman Emperors" on Wikipedia (https://en.wikipedia.org/wiki/List_of_Roman_emperors), which presumably uses the most commonly used names for the emperors in question, the naming of them does not appear to be consistent. For instance, the full names of both emperors Titus and Vespasian were the exact same (TITVS FLAVIVS CAESAR VESPASIANVS AVGVSTVS). Why is one called Titus and the other Vespasian? If it is to tell them apart, why are some other emperors (e.g. Gordian I, II and III and the various Leos and Constantines) numbered instead? Some emperors also have later nicknames in place of their actual names, such as Caligula and Caracalla, surely they would have not used these names themselves. How were the emperors referred to formally by contemporary romans? Did they use the full name (e.g. TIBERIVS CLAVDIVS CAESAR AVGVSTVS GERMANICVS) or just the title "Augustus", how would one emperor be differentiated from another in historical texts?
There are several avenues to approach your question.
The most obvious way to ask how Roman emperors were 'actually' called would be to look at what the official versions of their names were. In that context, it is useful to look at how the first of the emperors, Augustus, came to his offical name.
Augustus was born as Caius Octavius Cai filius, and when he got adopted by C. Iulius Caesar, his name became Caius Iulius Caesar (Octavianus). By adoption, he had become part of Caesars family, the gens Iulia, and so took their nomen gentile, and also Caesars cognomen Caesar (though he did use it even before his adoption became official). By custom, he would have had Octavianus as a cognomen, since that was how an adopted Roman would usually signify their old family, but Augustus and his supporters never used that version, though Cicero does.
In October/November of 40 BC, after the treaty of Brundisium, his name became Imperator Caesar Divi filius. Imperator was actually more of a title, signifying the holder of imperium, or supreme command, but with Augustus, it became his praenomen, and would become standard part of the later imperial standard titulature. 'Divi Filius' simply means 'son of the divine one', since C. Iulius Caesar had by now been made into a god, so this was the way to refer to his divine parentage.
On January 16, 27 BC, he got rewarded by the senate with the title Augustus, which he took on as a cognomen. From that point on, his official name was Imperator Caesar Divi filius Augustus. Tiberius would inherit this title. Imperator, Caesar and Augustus oscillate between names and titles (if there is a significant difference to make in the first place, at least there is much overlap). All became part of the standard imperial title with time. Every emperor was called or called himself imperator (Tiberius f.e. was hesitant, as with the taking over of the Augustus-title, and was still known as Tiberius Caesar Augustus), but it still retained qualities of a title. The same is true with Caesar, which became a standard part of the name of the Emperor when Claudius and Nero, who weren't members of the gens Iulia took on this cognomen. Later, it became also somewhat of a title for the designated successor, when Galba gave the name to his adoptive son Piso (who never became emperor because of being murdered). It became somwhat of a first step on the way to the title of Augustus, which could also be awarded to a successor while the Emperor still lived, as was the case with Commodus, though they were still minor 'Augusti' without the full power of the original one, which only changed much later with Diocletian when the other Augustus had the full imperial bureaucracy and authority for his part of the empire to work with. In any case, what's important that these names begin to fossilize into titles through custom and inheritance, and Imperator really shouldn't be considered a praenomen after, say, Nero.
These official names usually appear in that form on official communication, such as building inscriptions, milestones (which customarily named the reigning emperor and thus served a dual function of honoring the emperor and counting the miles), honorific inscriptions for the emperor, often emended with titles and other names referring to victorious campaigns such as Germanicus, Arabicus, Britannicus, Sarmaticus and so on. What is name and what is title becomes a bit blurry at that point, since in later times they take on more of a character of an honorific title, and while Nero Claudius Germanicus was often and is usually today referred to as Germanicus, this is not the case for Marc Aurel (Imperator Caesar Marcus Aurelius Antoninus Augustus Armeniacus Medicus Parthicus maximus Germanicus Sarmaticus). As a complicating factor, emperors often would change their name, upon adoption, upon becoming Caesar or Augustus or for political reason (as with Commodus).
What an emperor was officially called is thus usually pretty straightforward to answer, so Marc Aurel is Marcus Aurelius Antoninus Augustus, depending on how many titles you want to add (in edge cases there can be discussion about what titles/names were officially in use, or at what point in time), but there is no hard an fast rule for how their were called by their contemporaries or by history, this is were it becomes a bit arbitrary. Romans usually referred to each other using their nomen, somthing like a family name, or cognomen, something which the term 'nickname' only loosely fits. See also this thread on how Roman names worked as an introduction if you want some clarification. In any case, that's why Caesar is called Caesar, Cicero Cicero (though some rhetoricians refer to him as 'Tullius' and some english historians as 'Tully'), Augustus is similar since it takes the place of the cognomen in his nomenclature. This holds true for lots of emperors, and most emperors are referred to by their one or more of their cognomina:
In some cases they are referred to by their cognomen and an ordinal number, usually inside of a dynasty or differentiate two rulers with the same cognomen:
The Historia Augusta already refers to 'the three Gordians', so that might be one reason for them getting associated with the numbers. Some are referred to by their nomen, which is also a common way to address another Roman:
Some are commonly referred to by a combination of the two:
Then there are Tiberius and Titus, who are referred to by their praenomen. In Tiberius case probably because that was the most obvious departure from the nomenclature of both Caesar and Augustus, with Titus this might be because he was called the same as his father and you had to differentiate the two. After all, that's the primary use of a name.
Some are referred to by nicknames they didn't commonly use during their time of reign, but which were commonly used around their reign. It seems to be no coincidence that all three are emperors that weren't treated too kindly by historical judgement.
Others are referred to with their cognomen and an ethnic qualifier signifying their origin or a peculiarity of their person, like Maximinus the Thracian, Philippus the Arab, Constantius the Pale, probably because this was seen as a significant differentiator.
or a mix of the two, here:
As you can see, using the cognomen is really the norm and there aren't too many that fall too far outside of that schema (I'm not going much further into late antiquity, because during the 3rd century, the praenomen all but disappears, and the nomen in time, and Romans tend to stop having multiple names - and emperors and officials all tend to be named Flavius, which becomes more of a title than a name).
Why some of them have numbers or other qualifiers added to their name is really quite arbitrary and a question of preference of contemporary or modern historians. Contemporary historians tend to use the cognomen as well, but of course they also differentiate them if it's not apparent from the context in which they appear. Suetonius for example calls Tiberius Tiberius Caesar when first introducing him, and Tiberius from then on, but it's immediately clear from the context that he isn't talking about just any Tiberius (just as when we talk about how Bush won the presidency in 2000 you know it's George W. Bush, not George H.W. or Jeb or really any other Bush).
We can also see how they were addressed by their contemporaries from some primary sources, for example Pliny's letters to the emperor Trajan, which usually start like this:
C. Plinius Traiano Imperatori,
Caius Plinius to the Emperor Trajan, using Trajans cognomen. Throughout the letters he also addresses him by terms such as domine, lord, or optime imperator, best emperor. Caracalla was referred to as Caracalla by his contemporaries soon after his death, as Cassius Dio relates (in 97,9,3):
But in general, much evil was continually spoken of him by everybody; in fact, people no longer called him Antoninus, but some called him Bassianus, his original name, others Caracallus, as I have stated, and yet others Tarautas, from the nickname of a gladiator who was most insignificant and ugly in appearance and most reckless and bloodthirsty in spirit.
There's a bit about Roman names here - https://www.reddit.com/r/AskHistorians/comments/1tegpu/roman_names/ , though more general than just the emperors.
In addition, some of those names were common for a reason - Caesar and Augustus were titles, Flavius & Aurelius are described in this askhistorian answer as becoming almost like a title.