In the Middle Ages there was a conflict between the Emperor and the Pope that was known as the Investiture Struggle. What was the nature of this conflict between church and state? It concentrates on the papacy and the papal claims to universal authority.
Investiture is defined as the ceremony or act of investing or installing someone in high office. Having that understanding it is easy to see why the question would be asked by the emperor, on what basis and by what authority, did the Bishop of Rome claim power over all other bishops and indeed over all Christians ? Some of the claim was based on biblical passages, but some of it derived from political and cultural realities.
Until Constantine moved the seat of power in the 4the century Rome was the most important city in the western world, suffice it to say being the bishop of an important city such as Rome would certainly gave increased stature to the holder of that office. It certainly gave the Roman bishop automatic prestige in the western Mediterranean, where there were no other cities to rival it. In the east in the east it was another story altogether, rivals included Alexandria, Antioch, Jerusalem and Constantinople and the bishop of Constantinople would never did bow to Roman claims.
Tradition conferred upon Rome a certain amount of special prestige. After all Rome was the city of martyrs. The first persecutions, instigated by Nero, took place in Rome and the list of martyrs reads as a veritable hall of fame in Christendom. Ignatius, Ploycarp, Justin, Perpetua, Falicitas and Peter just to name a few. (Gonzalez 43, 46 & 84) In addition the claim could be made that heresy never took root in Rome which was in direct contrast to the east which was a hotbed of heresy and cities such as Alexandria were permanently tainted by this fact, especially since some of their own bishops had been leading figures in the heresies. The Bishops of Rome, and their flock, had generally remained constant. Prestige builds prestige, especially in the west, so Rome gradually became the arbiter of theology and church authority. By the 5th century, it was generally acknowledged that any bishop deposed by a local council could appeal his case to the bishop of Rome. Eastern bishops would avail themselves of this arbitration as well. However none of this would have supported papal claims later in the age when the city of Rome no longer commanded such automatic respect. But the bishop of Rome could always point to the Bible to buttress his claims.
In the Bible, there are passages in which Jesus gives some very specific instructions to Peter. One such passage is recorded in Matthew 16:18-19, it is the reply of Jesus to Peter when Peter acknowledged him as the Christ:
And I say to thee, thou art Peter and upon this rock I will build my Church and the gates of Hell shall not prevail against it. And I will give thee the keys of the kingdom...