To borrow a phrase from the great John Green, papal authority developed "the way you fall asleep: slowly, and then all at once."

There is a debate in the literature on this subject, called the Ullmann-Barraclough Debate, as to whether the rise of papal power was intentional or accidental; here I intend to offer a middle way.

The bishop of Rome has always held some degree of prestige in the Church, but as the capital of the Empire shifted to Constantinople, true authority, even over ecclesiastic matters, remained with the Emperor. Rome held some moral authority, both through its antiquity and through the personalities of the holders of the see, like Leo I (the Great).

With the fall of the western Empire in 476, the bishops of Rome took over administration of the city. The Empire maintained the nearby city of Ravenna as its base of power, but was unable (or uninterested) in providing any major military or financial support, and Rome became a (mostly) autonomous polity under episcopal authority. In the late sixth century, Gregory I (the Great), for example, was clearly the spiritual and temporal ruler of the city, distributing the grain dole which he funded himself and negotiating with the invading Lombards.

Freed from the direct threat of Imperial authority, Rome began to assert itself on issues of doctrine, opposing the claims of the emperors and/or the patriarchs of Constantinople to any sort of overarching authority on such matters. Indeed, the first use of the now-popular papal title servus servorum Dei (servant of the servants of God) was in the works of Gregory I as he argued against Constantinople's - or anyone else's - right to ecumenical (universal) authority over doctrine.

This first power struggle between Constantinople and Rome was pretty messy, with doctrines affirmed and rejected and Imperial troops kidnapping one or two occupants of the chair of St. Peter. When the Empire was strong (e.g. under Justinian I), Rome was subordinated to Imperial demands; when it wasn't, it wasn't. The important take-away here is that the prime concern of the Roman episcopate during the fifth through seventh centuries was ecclesiastical and doctrinal independence from Imperial authority, and the preservation of "orthodoxy" vs. the changeable, "heretical" theologies that came out of Constantinople.

The Exarchate of Ravenna finally fell to "barbarians" in 751, removing the last vestiges of Imperial power from the Italian peninsula. Enter Charlemagne. With the Carolingian conquest of the Lombards in the late eighth century, Rome once again had a major temporal power at its doorstep. When Charlemagne came to Rome in 800, he was either unexpectedly (if you supported Charlemagne) or deliberately asked (if you supported the Pope) to be crowned Emperor of the Romans by Leo III. Whichever version you believe, the act was a clear assertion of religious authority by the papacy, but it was also one that was undertaken with the previous conflict with the eastern Empire in mind. Leo III's action was a clear rejection of Constantinople AND an assertion of his own authority over that of Charlemagne.

The papacy in the ninth and tenth centuries was a mess. As Charlemagne's empire crumbled, the papacy became the plaything of the Roman nobility, with popes hunting, hawking, philandering, and poisoning (or being poisoned by) their enemies - in short, acting like secular princes. But in the tenth century, Otto I was able to (re)establish the Holy Roman Empire and began to go about the business of reforming the church. The Emperors began to intervene directly in papal elections just as the conflict with Constantinople reached its climax with the Great Schism.

This renewed meddling by the local temporal power had some pretty bad memories for the Roman church, which began to push back. A fight erupted between Pope Gregory VII and Henry IV over the right to invest bishops with authority, known as the Investiture Contest. Gregory clearly had previous conflicts of Church and Empire in mind, dusting off the title of servus servorum Dei, unused since Gregory I. He also began to make formal assertions of rights to appoint bishops, determine doctrine, and even depose emperors. How new such claims were is a subject of debate, and that is another post. The important piece is that here, the papacy began asserting a temporal authority beyond the boundaries of the city of Rome.

The Investiture Contest went on for several centuries, but for a variety of reasons, the Papacy held the upper hand. In the process, Rome accumulated more and more power, and became an ever more central authority. The Pope took the right to an increasing number of episcopal appointments and collected tithes from across Christendom. Canon law and ecclesiastical courts were regularized. All roads once again began to lead to Rome. And a papal confirmation that your local patron saint was really a saint began to gain prestige, leading to the development of a formal canonization process. By the pontificate of Innocent III (1198-1216), the Papacy had arrived in a recognizable form.

Hey! SheepExplosion has done a good job with your question, but I answered a similar one a couple of days ago. You might find it helpful as well :)

Well that's up for contention, as the Orthodox (all non-Catholics really) still don't believe in his universal patrimony. To understand the rise of the Papacy, you need to understand the split between the two major churches. Traditional historiography points to the Great Schism of 1054 as the breaking point between Catholicism and Orthodoxy, when a papal legate excommunicated Patriarch Cerularius, who in turn, excommunicated the legate (the ecclesiastical validity of this is questionable, considering that the legate, a Cardinal, did so after Pope Leo X died, and that Patriarch Cerularius only excommunicated the legation, not the Papacy as a whole). This was the straw that broke the camel's back, but the two churches had been feuding de facto for a while before. Let's explore this, shall we?

Backtrack a few centuries and we the 1st Nicene Council, one of the most important Ecumenical Councils in Christian history. It is here that Constantine first organized the Church (after decriminalizing it in Milan) into the major patriarchates (but not quite the pentarchy) and established numerous items of theological importance, like the eponymous Nicene Creed (still used today in conjunction with the older Apostolic Creed in Catholic Churches worldwide). The most important thing to take from this, however, is that the bishops at this synod ranked the patriarchal seats in order of importance. Rome was made supreme over others (though still in a primus inter pares, first among peers sense) alongside Antioch and Alexandria (plus Jerusalem). Constantinople is still under renovation as Nova Roma, and ole' Byzantium has yet to come into the picture. The Christian Church (though a solid 300+ years old now) was still in its formative stages and the first signs of conflict have yet to occur.

A little bit later, the First Council of Constantinople adds Constantinople as one of the great sees of Christianity, and each patriarch is granted specific geographic regions of authority (though Rome, Antioch, and Alexandria had been granted some previous dispensation of authority beyond their immediate province). Arianism is declared a heresy here, and there is a famously apocryphal of how Santa Claus (aka St. Nicholas of Myra) punched Arius in the face for being a heretic (Gustav Anrich's Hagios Nikolaos claims that this was an amusing historical exaggeration to improve the public image of the Council).

Now here comes the biggie: the Council of Chalcedon (451), where one of the first big schisms starts to occur. Chalcedon upholds the doctrine of the Trinity and another important idea: Jesus's complete divinity and complete humanity. This does not fly with many of the Eastern Oriental Churches, like Coptic, Armenian Apostolic, and Nestorian (and the other Miaphysite and Monophysite branches of Christianity who hold that Christ had one nature or a mixed one) broke off, and the Patriarch Dioscorus of Alexandria was kicked out for being non-Chalcedonian (the Patriarchate of Antioch would flip-flop for a while as Chalcedonians and non-Chalcedonians vied for imperial favor). Most importantly in this council, in a canon of disputed validity, Constantinople is named second in eminence to Rome, setting the stage for the Rome-Constantinople rivalry that would dominate Chalcedonian ecclessiastical politics for centuries to come. The Bishop of Rome was startled by the idea that Antioch and Alexandria were essentially demoted, and thus filed a complaint against this canon, Canon 28. Tensions start to rise, but are defused for the time being as Rome, Constantinople, Jerusalem, and the new Patriarchs of Alexandria and Antioch remain united.

Now we get to what is possibly the most important factor here: the collapse of Western Roman administration. Odoacer deposes the usurper Romulus Augustus, and in a daring move, doesn't bother to appoint a puppet emperor, unlike other maior domos before him. Emperor Zenon of the East eventually gets irritated at Odoacer and allows Theoderich and his Ostrogoths to invade Italy. Biggest consequence here: Rome is no longer under the temporal authority of the Roman empire, while Constantinople, Antioch, Jerusalem, and Alexandria still are.

This is huge.

The Western part of the Empire still relied on Latin as a primary means of communication throughout the diverse parts of the Empire, supplanting the various Celtic and Iberian languages spoken throughout the provinces. The Eastern part, however, retained a strong tradition of Hellenistic literature and customs, and so Greek was the active lingua franca of the Eastern Mediterranean (as it had been for centuries, hence why some Biblical historians claim that Jesus was fluent in Greek as well as Aramaean). While the official language of the Empire would only switch to Greek under Heraclius, the liturgical language of the East was done by and large in Greek, while the West preferred Latin. With the loss of the Western provinces, it became increasingly harder to find people who could speak both Latin and Greek, so communication between East and West became more difficult. With a linguistic split came numerous minor splits between the Latin and Greek rites, such as the use of leavened (with yeast, symbolizes the rising of the soul through the bread, used in the East) and unleavened (what Jesus used in Passover, used in the West) bread, priestly celibacy, how to represent Jesus in iconography, etc.

While Justinian I was able to briefly regain authority over Rome, the rule of his successors would put Greeks in the papal seat, which infuriated many Latins, who, as Herwig Wolfram put it in his History of the Goths, preferred the "tolerance of the Arian Gothic King to the caesaropapist claims of Constantinople".

The Arab Invasions sweep in during the 7th century, coinciding with the Heraclian imposition of Greek as language of the West. Not only did this deepen the linguistic and cultural divide of the Christian church, it also took 3 of the 5 Patriarchates set up by Justinian I out of the picture; Alexandria and Jerusalem would be gone from Roman authority forever, and Antioch would only be briefly held before once again being lost to the tides of Islam. This is important as it leaves only two Patriarchs left with any sort of temporal power: the Patriarch of Rome, outside of Roman authority, and the Patriarch of Constantinople, whose power was intrinsically tied to that of the emperor.

The Patriarch of Rome, henceforth the Pope, would vie for preeminence with the Patriarch of Constantinople, and while a number of pretexts have been used to explain sour relations between the two (dispute over the Filioque clause, leavening your bread, etc.), the problem at its core remained political. Proselytizing in the West was almost exclusively the domain of the Latin Rite of Christianity, so as new peoples were brought into the fold (i.e. the Franks, Saxons, Norse, Wends) they were indoctrinated under the Latin Rite. The Papal coronation of Charlemagne as Holy Roman Emperor in 800 established the Pope as the supreme ecclesiastical authority of Western Europe. The key thing to note about the relationship here is that the Holy Roman Emperor relied on the Pope for his legitimacy but did not consider him subservient to the state, whereas in the East, caesaropapist tradition restricted the Ecumenical Patriarch to a subservient position, even if he did crown the new Eastern Roman Emperors. Unlike in the West, when new kingdoms were carved that followed the Greek rite, caesaropapism kept the local autocephalous patriarchs leashed to their temporal authorities (as in Bulgaria); not only did the "first among equals" principle preclude any one Patriarch from exerting full ecclesiastical authority over the others, but the geopolitical realities of the time (the strength of each temporal state) also determined if and when a patriarchal authority could be projected.

So here we are once again at 1054, and the Ecumenical Patriarch and the papal legation have just excommunicated themselves. The churches at the time didn't really consider it a big deal, but the political realities of the world that followed (the Seljuk invasion, the crusades) essentially sealed both the Catholic and Orthodox churches' fates. The Latin Massacre and the 4th Crusade soured relations between both churches permanently, and the fall of the Eastern Roman Empire to the Ottomans (and the subsequent rise of authority of the Autocephalous Patriarch of Moscow as the temporally preeminent Orthodox cleric) meant that the Ecumenical Patriarch was essentially doomed to be the ethnarch of the Greeks under Ottoman rule. Meanwhile, the rise of Ultramontanism and the absence of caesaropapism in the Latin rite allowed the Pope to accumulate unprecedented temporal power and political autonomy, and the political ascendancy of the Iberian states (whose national and ethnic identities are deeply rooted in Catholicism, given their centuries-long war against the taifas of Andalusia) on the world stage ensured that papal authority spanned the globe.

TL;DR: the Pope had always been considered "first of equals" as determined by the early church councils, but it is only when the church split and Rome was not leashed by caesaropapism and imperial authority that the Pope was able to exert tremendous political and ecclesiastical power, taking his role from just "the successor of St. Peter" to "Christ's vicar on Earth" and giving rise to dogmas like papal infallibility. The success of Catholic states like Spain, Portugal, and France in their colonial conquests helped spread Catholicism throughout the world. Meanwhile, the Ecumenical Patriarch was constrained by his subordination to the Eastern Roman Emperor and thus had to cede ecclesiastical authority to autocephalous patriarchs as other Orthodox states eclipsed a dying Byzantium in power.