The Tetrarchy in Roman history refers to the division of the Roman Empire into a western and eastern empire, with subordinate divisions within the western and eastern empires.
Tetrarchy comes from the Greek words for four (tetra-) and rule (arch-) or what could be called a quadrumvirate (4-man [rule]) if basing it on Latin, as would seem more apporopriate for a Roman system of rule.
Tetrarchy refers to the establishment by the Roman Emperor Diocletian, in 293, of a 4-part division of the empire. Diocletian continued to rule in the east. He made Maximian his equal and co-emperor in the west. They were each called Augustus which signified that they were emperors. Subordinate to them were the two Caesars: Galerius, in the east, and Constantius in the west. An Augustus was always emperor. Sometimes the Caesars were also referred to as emperors.
This method of creating emperors and their successors bypassed the need for approval of emperors by the Senate and blocked the power of the military to elevate their popular generals to the purple. [Source: "The City of Rome in late imperial ideology: The Tetrarchs, Maxentius, and Constantine," by Olivier Hekster, from Mediterraneo Antico 1999.]
The reforms of Diocletian came after a period when many emperors had been assassinated. This earlier period is referred to as chaotic and the reforms were meant to remedy the mess that the Roman Empire faced. With 4 men to move through, assassination became less likely, so at least in this sense, this was a more stable form of government.