So, obviously, the first major event for Christianity in Late Antiquity was the Edict of Milan in 313 CE. This was a proclamation of religious tolerance by Constantine, issued jointly with the Eastern Emperor Licinius. "To the Christians and all men we decree that there be given free power to follow whatever religion each man chooses so that, whatever gods there be, they may be moved mercifully towards us." (There had been, of course, previous edicts by Constantine and his predecessor, Galerius, to halt the persecution of Christians). Over the next centuries, the Church became more closely tied with the Empire (with the curious exception of Julian the Apostate). This official recognition gave Christians an important power: bishops could serve as judges in civil disputes involving Christians. The church also developed a clear hierarchy based on imperial administration. Bishoprics were based around Roman cities since the days of the early church, but now provinces were organized into archbishoprics. Bishops of four great cities came to the fore: Rome, Jerusalem, Alexandria, and Antioch (Constantinople had not achieved prominence yet and would join these four later in the 4th Century).
The conversion of the emperors was not without incident, however. Constantine's successor and son, Constantius II was met with hostility when he called church councils at Arles and Milan. Bishop Hosius of Cordoba wrote, "You have no power in the ministry of sacred things." The Bishop of Poitiers called him "Constantius the anti-Christ." While not the Pope, Bishop Ambrose of Milan, then the seat of imperial authority in the West, coerced Emperor Theodosius I to do public penance for a massacre he ordered at Thessalonica. This is a prime example of the church flexing its muscles, so to speak, over the empire.
As for the Bishop of Rome, specifically, they held an important position. Rome was not only a large city, but also vastly important as the church of St. Peter, thereby making its bishops the successors of St. Peter. The first explicit acknowledgement of the Bishop of Rome's power came in 344, when the Council of Sardica decreed that any bishop deposed by a local council could appeal his case to the Bishop of Rome for a final judgment. It is also important to note that many Bishops of Rome from the fifth century, including Leo the Great, came from old patrician families and brought with them a tradition of authority that went back centuries. Perhaps Leo's greatest success came with the Council of Chalcedon 451, where the bishop of Rome was reestablished as primate of the Christian world. Importantly, the Bishop of Rome was the only important bishop in the West while the East had four competing bishops in Alexandria, Antioch, Jerusalem, and Constantinople. Constantinople usually held the special position among those, and definitively so once the others were conquered by Muslims.
Later, as the emperors increasingly regarded themselves as sacred rulers, popes claimed religious superiority. Pope Gelasius I wrote to the Byzantine Emperor Anastasius in 492, and in his letter he claimed, "the responsibility of the priests is more weighty insofar as they will answer for the kings of men themselves at the divine judgment." Justinian's invasion of Italy in the sixth century complicated things, as the pope was now an imperial appointee. This was eventually stopped by the Lombard invasions, but now the papacy was in danger of being dominated by "barbarians." Of course, this was averted by the emergence of Pope Gregory I, the Great, in 590. Gregory negotiated peace with the Lombards and exerted firm control over the Italian churches. He used his significant influence outside of Italy as well, maintaining correspondence with the rulers of the Franks, the leading bishop of Spain, and the Bishop of Constantinople. He also dispatched a mission to convert the Anglo-Saxons of England to Christianity, which proved very successful eventually.
I will end my discussion by mentioning Charlemagne. On Christmas Day, 800, Pope Leo III produced a crown and placed it on Charlemagne's head. Those assembled acclaimed Charlemagne emperor. This had long been a practice in the Eastern Empire, and anointment had been practiced at least since the days of Visigothic Spain. Charlemagne's realm was often called the "Christian Empire." The emperor and the pope, from Charlemagne on, had a mutually beneficial relationship for centuries.
Sources: Brian Tierney's Western Europe in the Middle Ages, as well as his Sources of Medieval History