The Latin Church was distinguished by the at least nominal leadership of the papacy based in Rome - indeed, it was the papal claim to leadership of the Christian Church as a whole that drove a permanent wedge between the western and eastern Churches, since the Byzantine emperors claimed authority over both Church and state. The popes were not just at the apex of the western Church, they often ruled as kings unto themselves, and they always had complex relationships with other rulers. For the entire period of the early Middle Ages (from the end of the western Roman Empire until the eleventh century), the popes were rarely acknowledged as the sovereigns of the Church outside of Italy. Instead, this period was important in the longer history of institutional Christianity because many popes at least claimed authority over doctrine and organization - centuries later, popes would look back on the claims of their predecessors as “proof” that the papacy had always been in charge.
An important example of an early pope who created such a precedent is Gregory the Great, who was pope at the turn of the seventh century. Gregory still considered Rome part of the Byzantine Empire, but by that time Byzantium could not afford troops to help defend the city of Rome, and he was keenly interested in developing papal independence. As a result, Gregory shrewdly played different Germanic kings off against each other and used his spiritual authority to gain their trust and support. He sent missionaries into the semi-barbaric lands outside of the kingdoms to spread Christianity, both out of a genuine desire to save souls and a pragmatic desire to see wider influence for the Church.
Gregory’s authority was not based on military power, nor did most Christians at the time assume that the pope of Rome (all bishops were then called “pope,” meaning simply “father”) was the spiritual head of the entire Church. Instead, popes like Gregory slowly but surely asserted their authority by creating mutually-beneficial relationships with kings and by overseeing the expansion of Christian missionary work. In the eighth century, the papacy produced a (forged, as it turned out) document known as the Donation of Constantine in which the Roman emperor Constantine supposedly granted authority over the western Roman Empire to the pope of Rome; that document was often cited by popes over the next several centuries as “proof” of their authority. Nevertheless, even powerful and assertive popes had to be realistic about the limits of their power, with many popes being deposed or even murdered in the midst of political turmoil.
Thus, Christianity spread not because of an all-powerful, highly centralized institution, but because of the flexibility and pragmatism of missionaries and the support of secular rulers (the Franks, considered below, were critical in this regard). All across Europe, missionaries had official instructions not to battle pagan religious practice, but to subtly reshape it. It was less important that pagans understood the nuances of Christianity and more important that they accepted its essential truth. All manner of "pagan" practices, words, and traditions survive into the present thanks to the crossover between Christianity and old pagan practices, including the names of the days of the week in English (Wednesday is Odin's, or Wotan's, day, Thursday is Thor's day, etc). and the word “Easter” itself, from the Norse goddess of the spring and fertility named Eostre.
As an example, in a letter to one of the major early English Christian leaders (later a saint), Bede, Pope Gregory advised Bede and his followers not to tear down pagan temples, but to consecrate and reuse them. Likewise, the existing pagan days of sacrifice were to be rededicated to God and the saints. Clearly, the priority was not an attempted purge of pagan culture, but instead the introduction of Christianity in a way that could more easily truly take root. Monks sometimes squabbled about the nuances of worship, but the key development was simply the spread of Christianity and the growing influence of the Church.