By Nino Barbieri - Own work, Public Domain, https://commons.wikimedia.org/w/index.php?curid=1697833
In the late Roman Empire, Diocletian divided the rule of the empire between two emperors known as augusti (augustus as the singular title) and two caesares (caesar as the singular title) who would be their juniors and successors, a system now known as a tetrarchy (rule of four). This was a reaction to the Third Century Crisis, in which Rome had faced a number of civil wars and foreign invasions which led to economic disintegration, nearly leading to the collapse of the Roman Empire.
Diocletian began the move to a tetrarchy with a diarchy, or rule of two, by assigning Maximian, also known as Herculius, as his caesar in 285. In 286, Maximian was promoted to augustus and the empire was divided between the eastern and western regions, with Diocletian ruled over, while Maximian reigned over the western portion. They reigned together as a diarchy until 293 when Diocletian decided that there needed to be focus on both civil and military problems, so he and Maximian expanded the imperial college by appointing two caesares, Gaius Galerius Valerius Maximianus to serve under Diocletian and Marcus Flavius Valerius Constantius to serve under Maximian. This became known as the first Tetrarchy. In 305, Diocletian and Maximian retired, elevating Galerius and Constantius to augusti and then they appointed Maximinus and Severus II, respectively, as caesares for the second Tetrarchy.
By Coppermine Photo Gallery - CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=4574473
While Rome was still considered the heart of the empire, the ruling centers were moved to where it was easier to defend the empire from its enemies, especially Sassanian Persia and the barbarian tribes of Germania and nomadic and displaced tribes in the eastern steppes. The four capitals were Nicomedia, near modern day İzmit, Turkey, which served to defend against the Persians; Sirmium, near modern day Sremska Mitrovica, Serbia on the Danube, which became pivotal in the defense of the Balkans-Danube region; Mediolanum, near modern day Milan, with easy access to the Alps; and Augusta Treverorum, near modern day Trier, Germany, which was close to the Rhine border. Two other cities, Aquieia on the Adriatic cost, and Eboracum, modern day York, England, also were important centers of defense for Maximian and Constantius respectively. There were no exact borders of the tetrarchs influence and the state wasn't divided into sub-empires, but it was more that they had their 'war theater' of command, as they spent a lot of time on the battle field and left the day-to-day bureaucracy to praetorian prefects.
While there might have been strife between the tetrarchs, the public image was carefully controlled to present a unified front, to increase the confidence of the citizens of the empire. This was furthered by the tetrarchs all having identical official portraits on coinage, with only the inscription identifying which emperor it is meant to be depicting and their statues being made identical, as well, even to the same military costume.
In 306, Constantinus died and Constantine was declared augustus by his father's troops but Galerius decided to promote Severus instead and naming Constantine as caesar instead, leading to two augusti. Maxentius, Maximian's son, began to resent being left out of the chain of succession and defeated Severus, forcing him to abdicate, then causing him to be murdering him in 307.Maxentius and Maximian then declared themselves augusti. By 308, there were four augusti claimants and only one caesar. Galerius called an imperial conference, requesting that Diocletain and Maximian attend. The council decided that Licinius would be the western augustus and Constantine would be his caesar and in the East, Galerius would continue in his position with Maximinus continuing as his caesar. Maxentius, however, became the de facto ruler of Africa and Italy even though he held no imperial status. Additionally, Constantine and Maximinus resented Licinius advancing over them even though they'd been caesar since 306 and 305 respectively. In a bid to try to placate them, they were granted the empty title filius augusti (son of the augustus), they were then promoted to augusti in 309 with all four tetrarchs now augusti.
By School of Raphael - Constantine at the Battle of Milvian Bridge, Public Domain, https://commons.wikimedia.org/w/index.php?curid=679484
The next four years saw great upheaval in the imperial office. Constantine forced Maximian to commit suicide in 310 then Galerius dying of natural causes in 311. Maxentius and Constantine fought and Maxentius being finally defeated and killed at the Battle of the Milvian Bridge in 312. In 313, Maximinus committed suicide after Licinius defeated him in battle at Tarsus. This left two rulers, Constantine over the Western Roman Empire and Licinius over the Eastern Roman Empire. In 324, Constantine would reunite the Roman Empire as the sole augustus.
After the Tetrarchy, the Roman Empire continued to be divided into four, but the parts were administered by praetorian prefects, which were then divided into dioceses assigned to magister militum. The position of caesar would be intermittently used. The Roman empire would eventually splinter after the death of Theodosius I in 395, when the two halves would officially be ruled as a unit until the Western Roman Empire fell in the 5th century.
