Short answers, pulled from the story.
The Tetrarchy was a system of government instituted by Emperor Diocletian that divided rule of the Roman Empire among four rulers: two senior emperors called augusti and two junior emperors called caesares. It operated between AD 293 and approximately 324.
The four original tetrarchs were Diocletian and Maximian as the two augusti, and Galerius and Constantius I as the two caesares, appointed in March 293. Diocletian and Maximian governed the eastern and western provinces respectively, with each caesar answering to one augustus.
The four tetrarchic capitals, Nicomedia, Sirmium, Mediolanum, and Augusta Treverorum, were chosen as military headquarters near the frontiers, not as ceremonial centers. Rome remained the nominal capital of the empire but was no longer an operational base.
The system broke down after Constantius died in July 306 and his troops proclaimed his son Constantine augustus, bypassing the established succession rules. A cascade of rival claimants followed; by 308 four men held or contested the title of augustus simultaneously. Constantine defeated the last rival, Licinius, at the Battle of Chrysopolis, becoming sole ruler by around 324.
The term was first applied to Diocletian's system in 1887 by schoolmaster Hermann Schiller in a two-volume handbook on the Roman Empire titled Geschichte der Romischen Kaiserzeit. It did not enter widespread scholarly use until Otto Seeck adopted it in 1897.
Galerius defeated the Persian king Narseh in 298 after an earlier setback, capturing members of the Persian imperial household and securing a peace that held for a generation. Constantius defeated the British usurper Allectus, Maximian pacified the Gauls, and Diocletian crushed the revolt of Domitianus in Egypt.