The Roman Catholic hierarchy was not instituted by the apostles but rather it evolved over time. It was not even conceived by the apostles as some sort of acorn flowering into a mature oak tree (John Henry Newman's and Vatican II's idea of development). When Jesus told Peter that "on this rock I will build my church" (Mt. 16:18), even if we grant that the "rock" is Peter, Jesus made the same promise of the authority of binding and loosing to the other apostles (Mt. 18:18, Jn. 20:23). There is nothing special about Peter except that he was the one chosen to show how slow and foolish the apostles were. In fact, despite the grace given to Peter, even after Pentecost and after years as a church leader, Peter still fell into error and needed to be rebuked publicly by Paul in Galatia (Gal. 2:11-14). And while Peter gave the sermon at Pentecost (Acts 2:14-36), it was James who presided over the Jerusalem council (Acts 15:12-21)
The idea of a Petrine chair of apostolic succession is historically and biblically without any foundation whatsoever. So how did the papal office came into being? It came into being slowly over centuries, built upon many choices the church made that slowly but surely deviated from the biblical norm of ecclesiastical governance.
The early church was a built upon the Presbyterian model. Elders were appointed in every church (Acts 14:23, Titus 1:5), and deacons were likewise elected into office (Acts 6:1-3). The Apostles, as the special office, slowly passed from the scene, leaving every church with a plurality of elders and deacons, yet connected to each other.
As the apostolic church became the early catholic church, bishops began to emerge. The first churches were founded in major cities like Jerusalem, Antioch, Rome, Athens, Corinth, Ephesus among others. But as the Church grew, churches began to be planted in minor cities and towns. The pastor of the church in the major cities began to take on a mentoring and leadership role over the smaller churches, and they became the bishops. Thus, the episcopal model of governance began purely as a matter of good, helpful and efficient practice. Over time, power and influence began to converge into five major churches in the cities of Rome, Jerusalem, Alexandria, Constantinople (New Rome), and Antioch. The bishops in those five cities began to be seen as in some sense superior to the others, as they claimed apostolic succession for their respective sees.
While Presbyterianism has ceased to be practiced in the early catholic church, the equality of bishops persisted for a longer time. At Nicea in 325AD, the bishop of Rome was not seen in a special light. In fact, Liberius in his capacity of bishop of Rome (352-366AD) condemned Athanasius on behalf of the Arians, and was in turn anathemized by Hilary of Poitiers (Hilary of Poitiers, "Liberius, to the Eastern Presbyters and Fellow Bishops," Book II, VII; "Letter of Liberius in Exile to Urascius, Valens and Germinius," Book II, IX.). The bishop of Rome in the fourth century was taken to be just one of many bishops. He was the bishop of the capital city of Rome, but that did not come with any special privileges.
As the age of the united Roman Empire drew to a close, chaos began to spread in society. Over time, the Empire became divided over the issue of language. The Western half of the Empire began to speak exclusively Latin, while the Eastern half spoke Greek. The coming of the "barbarians," the Goths, Huns and other non-Romans who attacked and bled the Roman Empire, created space for the spread of the influence of the church. As society collapsed in the West, the Church stepped in to provide services for society, which is certainly a good thing yet it had unintended consequences. East and West moved further and further apart over language and culture and even jurisdiction. The vacuum of power in the West led the Church to began to assume secular power for herself, creating the two swords doctrine. In the East, the Emperor controlled the church, resulting in caeseropapalism, the emperor (Caesar) as "pope."
As the "apostolic sees" emerge, the various Patriarchs still embraced an equality among them. Unfortunately for the Church, only one of the apostolic sees was in the West - Rome. The bishop of Rome, sitting in a land without any equals, and sitting among the vacuum of sociopolitical power, began to be elevated in power and stature. Despite the first real pope Gregory the Great denying the use of the appellation "universal" to any one single bishop, in time his successors would embrace the term, amassing power and wealth into the see of Rome. By the time of the High Middle Ages, the papal office as the supreme leader of Christendom was established. Of course, the history of the church in the setting up of church councils was not forgotten, so some people still held that councils are superior to the pope, a movement known as Conciliarism. The fiasco known as the Babylonian Captivity of the Church, where a series of popes and anti-popes ruled and anathemized each other, became an utter disgrace to the Church and strengthen the forces of Conciliarism. During the Medieval period, the pope did not have absolute power over all of the Church, but must rule with the help of bishops and other members of the clergy. The Pope of course was very powerful, but he did not yet have the power of an absolute monarch although he certainly acted the part.
This situation persisted through the Reformation. As powerful as the Pope was, if more bishops had turned on him during the Reformation, he would be forced to try to find a middle ground with the Protestants. But the Pope had enough powerful people who sided with him, and many people did not like being mocked and ridiculed by Martin Luther, an "uncultured" German. During the Jansenist controversy after the Reformation, and especially later in the Ultramontanist controversy, the various warring factions began to see Rome as the court of final appeal.
Ultramontanism makes for an interesting prequel to the next evolution of the papal office. As the Enlightenment dawned, the advances in learning split the French clergy between the smart and powerful who attend elite schools and were trained in the latest learning, and the poorly-educated priests who were trained in the old ways and used as fodder to fill pulpits. The lowly priests resented their smarter elite cardinals and bishops, and appeal to Rome was used to take the elite French clergy (Gallicans) down. France was of course the most advanced Roman Catholic country at that time, while the other Roman Catholic countries were just beginning to face the Enlightenment. The tremendous changes caused by the Enlightenment frightened the pope, who had a taste of is power when Napoleon humiliated one of his predecessors, Pius VI, by attacking the Vatican and taking him prisoner in the late 18th century. The pope thus had lots of reasons to fear the Enlightenment, and Ultramontanism dovetailed nicely with his mood as the pope was turned to as the defender of the faith.
In a reaction to the Enlightenment, Pius IX in 1864 published his Syllabus of Errors as a rejection of "Modernism." This dovetails nicely into the next evolution of the papal office into one of infallibility, as a safeguard against "modernism" in any form. Vatican I occurred from the years 1869 to 1870, and Pius IX pushed through the novel doctrine of papal infallibility, ignoring objections from learned Roman Catholic historians like Ignaz von Dollinger. Standing as THE defender against modernism, the pope gained support from the Ultramontanes and passed the encyclical Pastor Aeternus defining papal infallibility as dogma. Ironically, at the end of the council before it could be formally closed, the Italian nationalists invaded the Vatican, stripping the pope of his secular powers just as he claimed supreme spiritual powers for himself.
Vatican II marks another shift in the papal institution towards something closer to the situation of the early medieval period. This did not happen however because the pope suddenly became humble and decided to give up his power. Rather, the Roman Catholic Church had grown too large for the type of central control to take place, and the Asian, American, African, and Latin America bishops revolted and forced changes that promote a more decentralized model. Apart from suspending the council altogether, the pope could do nothing but acquiesce and try to work out something that preserved some of his power while ceding parts of it to the other clergy. Post-Vatican II, the Pope has become more of a figure of authority than an actual monarch. The Pope's absolute authority promulgated in Pastor Aeternus still remains on paper, but he has become more of a paper tiger than a real tiger in these modern times.
Thus marks the evolution of the papal institution. It began with helpful practices, mixed in with some heterodox teaching, and the special environment of its time cultivated the papacy. Good and helpful practices over time may result in error, and nowhere is this more apparent than in the evolution of the papacy.
 Lionel Wickham, trans., Hilary of Poitiers, Conflicts of Conscience and Law in the Fourth Century Church (Liverpool: Liverpool University, 1977), 77-9. Translated Texts for Historians, Volume 25, Against Valens and Uracius: The Extant Fragments, Together with His Letter to the Emperor Constantius. Translated from A. Feder (CSEL), ed., Collectanea Antiariana Parisina (including Lber ad Constantium Imperatorem and Liber II ad (or con.) Constantium) (1916). As cited in William Webster, Holy Scripture: The Ground and Pillar of our Faith (Volume II: An Historical Defense of the Reformation Principle of Sola Scriptura; Battle Ground, WA: Christian Resources, 2001), 2:267