The crisis of the 3rd Century saw a very long series of very short-lived emperors for a number of reasons, defeats by the Persians in the east, difficulty raising enough taxes to support the army, and the sheer size of the empire. At a time with such primitive transportation and communication along with the increasing centralization of power in the hands of a single man, it was difficult for that man to maintain control over such a wide expanse. Increasingly he had to rule from the frontier near his armies rather than from Rome, and advancement in the administration often came to those in the presence of the man himself, thus if he set himself up on the eastern frontier, western generals tended to rebel and vice versa.

Diocletian (284-305), credited with a large number of reforms and bringing the crisis to an end (though many of these reforms at least began with those who preceded him), instituted the Tetrarchy, in which 2 senior "Augusti" and 2 junior "Caesars" ruled the empire. Each had his own lands, but 1 Augustus and 1 Caesar ruled the east, the others the west. The Caesars married the Augusti's daughters and each moved to Augustus after their respective senior emperors abdicated. This quickly fell apart when Constantine, son of the western Caesar, took over the entire empire during a long series of Civil Wars ending in 324.

From here on there is a long sort of transitionary period in which the empire is sometimes ruled by 2 men and sometimes by 1. Theodosius rules the entire empire from 392-395, and he is the last emperor to do so. His 2 sons succeed him, the elder in the east and the younger, Honorius with the help of a half-Vandal general Stilicho, in the west. It is under Honorius and Stilicho that the barbarian invasions of 405-6 happen and the western empire begins to fall. Given that Theodosius is the last emperor of both halves, we can probably think of this as the end of them as a single political entity.

They are still closely intertwined, however, and Theodosius's dynasty rules the east and the west (if we count inheritance through women) for the next 50 years. The eastern Theodosian emperors will continually support those in the west. I'm sure most people have a certain image of Roman generals whether of high birth or not taking the throne by force, and this definitely happened a lot during the end, but until the mid 450's those men could really only name themselves Magister Militem (master of soldiers) because they needed to keep the Theodosian emperor on the throne to ensure alliance with his eastern relative. The east would also attempt to help with 2 reconquests of North Africa and they would harry the Huns during their western invasions, so their ties were strong but they were 2 different entities by the end.

Sources: Peter Heather, The Fall of the Roman Empire: A new History of Rome and the Barbarians

Also, if you don't feel like reading there is this yale course on the early middle ages which begins with all this.