The language used to describe the Early Church can often be the biggest barrier to engaging helpfully with it.

Regardless of your theological persuasion, and your own thoughts on the polity (and autonomy) of the local church, the world of the Early Church can seem alien. Who led these first christian communities?
The answer to this is simple, bishops. The name comes from the Greek, ἐπίσκοπος, and literally means overseer or supervisor. Indeed, Paul uses the word many times, notably in his description of the overseers Timothy ought to select to help lead the church in 1 Timothy 3:2.
“Now the overseer is to be above reproach, faithful to his wife, temperate, self-controlled, respectable, hospitable, able to teach…”
1 Timothy 3:2
The bishops we see in this early period of church history are not then the lofty ceremonial positions we have today in many denominations. They were pastors, elders and church leaders. Some, such as those of Rome or Alexandria, began to develop significant power or authority, but more often than not they were humble figures leading small Christian communities.
Many examples of these early Christian leaders are available for us to study. One of them, bishop Ignatius of Antioch, wrote a letter to the church at Smyrna in the early second century. In it he wrapped up the idea of the bishop in the local and universal church.
“Wherever the bishop shall appear, there let the multitude [of the people] also be; even as, wherever Jesus Christ is, there is the catholic church.”
Ignatius of Antioch, Letter to the Smyrnaeans, 8.
Ignatius uses the language of bishops and catholic churches, and he is mirrors the Biblical picture of church order. This is not an affirmation of the Catholic tradition. Nor is it an affirmation of a Protestant structure of church polity (or any other denominational setup). It is a simple explanation of the status of each and every local church community. An overseer (or overseers), leading a body of believers, under the authority of Christ, as a local part of the wider Church of Christ. A beautiful picture of the church as ordained by Scripture. A simple picture of how these Early Churches were led. By faithful men, set apart for ministry, under the law and rule of God.
For more on Ignatius himself: have a read here.