The really short and quick answer is that the Papacy begins as bishop who claims special connection to St. Peter but keeps mostly to the affairs of Rome. In the 11th century, popes begin to assert themselves and what they see as their rights and authority over other bishops and ecclesiastical affairs at the expense of local kings.

Here is a longer, more detailed answer.

The papacy underwent a large number of changes during the middle ages. Most of the authority people assume the pope has always had did really begin until the latter half of the 11th century, and it wasn't until the end of the 12th that they were really firmly cemented.

For starters, during the early history of the papacy the pope was principally thought of as the bishop of Rome and not some sort of sovereign over all ecclesiastical affairs in Western Christendom. As with much of the politics of the time, church affairs were primarily local affairs. The pope was believed to be an important bishop, as not only did apostles Peter and Paul die in Rome, but there were also a large number of other saints and relics in the city as well. However, popes of this period did little to influence the affairs of the rest of western Christendom, and and If you read Gregory of Tours's History of the Franks, he mentions the pope very little, except at the end of his work when he comments on the future Gregory I and his desire to convert the British Isles (a large part of which was already christian, but that's another issue all together.)

Anyways, back to Gregory I. Gregory I, also known as Gregory the Great, was one of the first very active popes in the early middle ages. He as a prolific letter writer and we still have many of his letters. Unlike many popes before him, Gregory was very active in the affairs of Christendom across the Mediterranean. not in the sense that he made decrees and enforced rules. Rather, many people wrote to him with questions and for advice relating to church matters. As i already mentioned, Gregory sends a mission to convert the British Isles back to Christianity. His delegation succeeded, and Christianity gradually spreads across Britain from the south up to the extent of Irish Christianity which moves from the north down to about modern day York. One key difference between the Irish Christians and the new "roman" Christians at this time is that the roman Christians look to Rome and the pope for spiritual guidance and authority. All because it was Gregory I who sent that first mission.

Fast forward to the 11th century. A group of churchmen who many historians now refer to as the "reformers" or the "reform party" decide that the state of the church is no longer acceptable. They cite numerous flaws such as clerical marriage and a lack of monastic rigor. However, their biggest target is the practice of Simony, or buying and selling church office.

Before the reformers simony was considered something akin to modern political corruption and taking bribes. While it was never considered acceptable, it was none the less a very common practice. The reformers gain key positions of power within the church by allying themselves with the then Holy Roman Emperor Henry III, who places them into key positions withing the church in exchange for political support, including the papacy.

When Henry III dies his son Henry IV is a mere 3 years old. His regency and early reign contain a great deal of turmoil. During this time the reformers continue to advance their own agenda, principally denouncing simony. In the year 1073 a man named Hildebrand become Pope Gregory VII (it is not a coincidence that he chose Gregory). Gregory went a step further than the rest of his counterparts in denouncing simony. Not only was someone who bought his church office a simoniac, so too was the man who was made bishop by a king.

Rewind to Gregory of Tours. Gregory writes a lot about bishops and the drama involved in a bishops succession. He mentions a number of ways in which one can become bishop, and it often involves a king stepping in and proclaiming the new bishop. However, he never mentions the pope in any of these incidents. Gregory's claim that only the pope had the authority to invest bishops, and that any king or nobleman who gave a man a bishopric or who received it from him were simoniacs flew in the face of established methods of succeeding bishops for hundreds of years. Even many of the reformers received their office from Henry III which could have been call simony by Gregory VII.

When Henry IV came of age he tried to do as his father did and invest men of his choosing into church office and Gregory VII tried to stop him. The conflict between the two sparks the Investiture Conflict, or the conflict over who has the right to nominate new bishops, the king or the pope. To cut a very long story short, Henry ends up briefly submitting to Gregory, but reverse his position and invades Italy, partially with the intent of ousting Gregory as Pope.

The conflict is not resolved with either mans death, but the damage had already been done. The papacy had made a claim about it's authority over spiritual affairs that one could say lead to a very gradual separation of church and state. This is not even close to the whole story of the evolution of the papacy in the middle ages, but I think it answers your question in terms of its evolution through the 'dark ages'