I love this question! Really cool and important, and also very answerable. I can’t really speak for the period before Diocletian’s Tetrarchy began in 285. I believe the first Roman co-emperor was appointed by Marcus Aurelius, but I couldn’t tell you much more than what you could find out for yourself from Wikipedia. From the Tetrarchy onwards all co-emperors were divided by both seniority and geography. Diocletian established the Tetrarchy with the hope of restoring the integrity of the Empire’s defences. At the time, which I wrote about in a previous answer here, the main armies of the Roman Empire now travelled with the emperors themselves. So wherever the emperor was, the Empire was mostly stable, but wherever the emperor wasn’t, the Empire was weak and vulnerable. By first promoting Maximian, and then later Galerius and Constantius Chlorus, and having them create new permanent residences closer to the Empire’s vulnerable frontiers, Diocletian greatly restored the Empire’s sovereignty.

Each of these emperor had their own geographic location that they controlled, effectively dividing the Empire into four sections. Diocletian took the eastern frontier with Persia, Galerius took the Danube, Maximian had Italy as well as Spain and most of North Africa, and Constantius Chlorus took Gaul and Britain. Each emperor was the primary authority in their provinces. They commanded the armies there and oversaw building projects and defences. So this was the first division of powers and authority.

There was also a second division. There was a seniority hierarchy amongst the emperors. Diocletian had been proclaimed as emperor by the army in 284, something that had become common during the third century. In 285 he raised Maximian first as a Caesar, making him a junior emperor, something a bit like a prince. Only in 286 did he raise him to the full status of Augustus, technically the same rank as Diocletian. When they later raised Galerius and Constantius Chlorus, they were only raised to the ranks of Caesar until Diocletian and Maximian retired. However, Diocletian ensured that it was recognized by everyone that he had raised Maximian, he had been emperor longer, and he was the senior. This message was conveyed in a plethora of ways, from Diocletian’s name coming first on official documents to iconography and their images on coins. It was made very clear that Diocletian was the more senior emperor. By extension of this, Galerius, as Caesar under Diocletian, outranked Constantius Chlorus but was superseded by Maximian.

In the end, the idea behind the Tetrarchy was too radical and idealistic for the Roman culture, and it quickly collapsed and the Romans reverted back to hereditary dynastic succession. Constantine I wiped away the last vestiges of Tetrarchy in 324 when he made himself sole emperor. Throughout his reign he chose to raise four of his sons to the rank of Caesar, including Constantine II in 317 even though he was just one year old. These four sons, Constantine II, Constans, Constantius, and Dalmatius, were all Caesars at the time of Constantine’s death in 337. They often held their own military commands or commanded alongside Constantine himself prior to his death, but their rank of Caesar while Constantine was an Augustus made it clear that they were his juniors in authority.

After 337 the Roman Empire was divided amongst the four Caesars, who all became Augusti. There was no real discernable hierarchy amongst them even though Constantine II was the most senior, in age and imperial years. They would die one by one until Constantius was the sole remaining heir of Constantine in 350. Constantius would first appoint Gallus as Caesar and then execute him for disloyalty, and then later appoint Julian to the same junior imperial rank, who would rebel and march against Constantius. Gallus controlled the east while Constantius was in the west, and when he returned to the east Julian controlled the west. So there was still a geographical divide and an official seniority divide.

Julian would appoint no heirs (unless you count Procopius but that was probably just propaganda), and neither would Jovian, the army proclaiming successors after their deaths. Valens and Valentinian would be the next co-emperors. They were brothers, Valentinian being the older and the one first proclaimed by the army, but he did raise Valens to the position of full Augustus. Again, there was a geographical divide of Valentinian in the west and Valens in the east, and Valentinian being the senior imperial figure. When they were briefly together at the start of their rule all laws and commands were issues by Valentinian alone.

Valens also appointed no heir, but when Valentinian became very ill in 367, the army began trying to find a successor. When he returned to health, the army was still concerned about the well-being of the Empire, and demanded he nominate a successor. Rather than choose one of the senior and experienced candidates the army was leaning towards, he chose his eight-year old son, Gratian. But the crucial thing here is that Gratian was not raised to the junior position of Caesar, but as a full Augustus. This was a major development, because Gratian was too young to be sent away or given his own authority; he had to remain with Valentinian. So there was no official divide in titles, nor a geographic divide. There was a major divide in reality however: when Valentinian took Gratian on campaign in 368 the sources record that Valentinian personally conducted dangerous missions against the Alemanni enemies, while Gratian remained in the year of the camp under heavy guard. While true, the was probably emphasized by Valentinian to showcase his preeminence.

Valentinian would die in 375 while with the army on the Danube river, while Gratian was nearly 900 kilometers away in Gaul. The army on the Danube now panicked; Gratian was only sixteen and he was militarily inexperienced. The generals and officials did not know what to do. There was the major risk that the Quadi, just across the river, might take advantage of the situation to attack. There was the risk that if no successor was appointed by them, the army itself might proclaim a successor of their choosing, and they had chosen poorly in the past. Even if the generals chose someone they liked, there was the risk that Gratian, and his advisors, would not like the choice and potentially declare them a traitor and force a civil war. But if they waited to ask Gratian for his approval of a candidate, the Quadi or Roman soldiery might do something rash.

Fortunately, the magister militum Merobaudes, the supreme commander in the area, had a solution. They summoned Valentinianus, Valentinian’s four-year old son, who was very close, and raised him as full Augustus. Gratian was unable to disapprove of his half-brother, and allowed his ascension and even summoned him to his person so he could oversee his upbringing. At some unknown time, Valentinian II, as the new boy-emperor is known, was sent to Italy to oversee the peninsula, as well as Illyricum and North Africa. However no laws were issued by him in this time; everything of importance came from Gratian. Again, it is clear that Gratian held the more senior authority. He also had control over all of Valentinian II’s armies.

However once Gratian was killed in 383, Valentinian II experienced what has been termed by historians recently his ‘real ascension’, as opposed to his ‘phantom ascension’ in 375. With Gratian dead, Valentinian II was not the western Roman Empire’s senior emperor. However, Valentinian II would become to be dominated by Arbogast, a different magister militum. Arbogast had exerted control over the young emperor before he was mature and old enough to control affairs on his own, and even when Valentinian grew older, Arbogast refused to give up his power over the emperor. From this point onwards, with few exceptions, the generals of the Roman Empire exerted almost complete control over the Roman emperors in the west. The raising of young sons as Augusti continued; for example, Honorius raised his less-than-two-year old son, Valentinian III, as Augustus in 221. These would be continued to be governed by the same father-son seniority that Valentinian I had with Gratian, although the emperors would no longer actively campaign to demonstrate this seniority.

In the fifth century many generals would also temporarily be recognized as co-emperors alongside the ‘main’ line of emperors who resided in Italy. I am not entirely familiar with the complexities of this period, and it is extremely complex. These generals, for example Constantine III, would become powerful enough to be recognized as co-emperor, but would then be defeated by rivals and replaced. Give me a couple of years, because I hope to do my PhD studies around this topic, and then ask this question again!