Here's something I wrote years ago (I would change some things in it now if I were to edit it, but it'll do I suppose)...
What was leadership in the early Church like? Was it "every parish for itself?" Were they connected solely through some mystical, invisible tie as part of the body of Christ? Were there bishops (ie. hierarchs) who held authority? And if there were bishops, does this mean that the Roman Catholic claims about Peter must be true?
The first thing that needs to be considered is whether the Scripture speaks of people specially set aside for a leadership role. Jesus Himself picked twelve specific disciples to have a special place in the early Church (Matt. 10:1-4; 28:16-20; Mark 3:12-19; Lk. 6:13-16; 9:1-2; etc.), and he also picked three of those disciples to have an even more prominent role than the other nine (Matt. 17:1-9; 26:37; Mk. 5:37). At another time our Lord sent 70 specific disciples out (Lk. 10:1). Jesus clearly said that he had founded a Church (Matt. 16:18-19), and that Christians would be bound to follow the leaders of this Church (Matt. 18:17-18; Jn. 20:23).
Jesus told the Jews of his day that they should follow the teachings of even corrupt Pharisees (Matt. 23:1-3), so is it really such a stretch to believe that Jesus would establish a hierarchy and want Christians to follow their leaders? If Jesus didn't establish specific leaders that he expected to be followed, then a number of Scriptural verses are pointless. For example, "Remember them which have the rule over you, who have spoken unto you the word of God: whose faith follow, considering the end of their conversation" (Heb. 13:7); and "Obey them that have the rule over you, and submit yourselves: for they watch for your soul , as they that give account, that they may do it with joy, and not with grief: for that is unprofitable for you." (Heb. 13:17)
The apostles did nothing more than continue the hierarchal system that they had been given by Christ our God. They quickly replaced the fallen apostle Judas with another (Acts 1:15-26), and later on specifically chose seven deacons to minister to the Church (Acts 6:2-6). Everywhere that they traveled they appointed specific people to be the leaders (Tit. 1:5; 1 Tim. 3:10; etc.) St. Paul said explicitly: "For this cause left I thee in Crete, that thou shouldest set in order the things that are wanting, and ordain elders in every city, as I had appointed thee" (Tit. 1:5). Leadership in the Church was a divinely inspired calling executed through the human instruments God had chosen. Paul speaks of this divine ordering when he said: "And God set some in the church, first apostles, secondarily prophets, thirdly teachers" (1 Cor. 12:28).
Among the Apostles called by Jesus, Peter was the leader. Peter's leadership can be demonstrated in many ways. Peter was always the first Apostle mentioned in the listing of the twelve (Matt. 10:1-4; Mark 3:12-19; Lk. 6:13-16), was among the three who were something of an "inner circle," (Matt. 17:1-9; 26:37; Mk. 5:37) and seemed the boldest of these three (Matt. 17:1-9; Mk. 9:1-13). Many of the important exchanges seen in the Gospels took place between our Lord and Peter representing the Apostles (Matt. 16:13-28; Jn. 6:61-69; etc.) And whatever one's interpretation of Matt. 16:18-19, it is probably bordering on eisegesis to attempt to exclude the person of Peter entirely from what is going on in the passage; there is clear patristic evidence that Peter was, in conjunction with the faith he expressed, being directly spoken of here as a rock and foundation.
While Jesus gave the powers of the Church to all the apostles, they were first (and foremost) appointed to Peter. Also, it is only Peter who is specifically given the keys by our Lord in the Scripture (though all the Apostles jointly held them in actual fact). As we move into Acts, we see Peter taking the leading rule in the evangelization of the people and defense of the Gospel (Acts 2:14; 4:8; 5:29; etc.) St. Paul clearly saw Peter as the leader of the Apostles (Gal. 1:18-19; cf Gal. 2:7-9). When some were disputing with Paul over Gentile converts, Peter was one of the ones to stand up and defend St. Paul's ministry (Acts 15:5-11).
However, this primacy of honor that Peter held was not a right to supremely (or infallibly) rule the Church. Peter was the lead pastor on earth, who worked together with the other Christians to bring about God's will. But he was only one brick in the entire wall of Christianity. Thus St. Paul could say to the Ephesians: "Now, therefore, you are no longer strangers and foreigners, but fellow citizens with the saints and members of the household of God, having been built on the foundation of the apostles and prophets, Jesus Christ Himself being the chief cornerstone, in whom the whole building, being fitted together, grows into a holy temple in the Lord, in whom you also are being built together for a dwelling place of God in the Spirit." (Eph. 2:19-22)
The Jerusalem Council as recorded in Acts 15 is a perfect example of how the Church hierarchy works. Who was it that gathered, and for what purpose? Acts records: "Now the apostles and elders came together to consider this matter." (Acts 15:6) And who spoke at this council, and who finished the discussion? It was Peter, Paul, and Barnabas spoke (Acts 15:7-12), but James had the final word (Acts 15:13-21). And what happened after the speech of St. James? Who made the final decision as to what to do? It was "the apostles and elders, with the whole church" (Acts 15:22), and not St. Peter or any other bishop by himself. The Council sent a letter with this phrase: "For it seemed good to the Holy Spirit, and to us" (Acts 15:28), which perfectly articulates a proper Christian ecclesiology. No person, not even the leader of the Apostles, works alone, but all work together for the good of the Church, guided by the Holy Spirit; however, within this context of collegiality, the leaders are still expected (and indeed obligated) to lead.
Unfortunately, people do not always work together as they should, and strife enters the Church. This is why, writing in 96AD, St. Clement of Rome (who had lived, learnt, and ministered among the apostles) said:
"The apostles have preached the Gospel to us from the Lord Jesus Christ; Jesus Christ [has done so] from God. Christ therefore was sent forth by God, and the apostles by Christ. Both these appointments, then, were made in an orderly way, according to the will of God... Our apostles also knew, through our Lord Jesus Christ, and there would be strife on account of the office of the episcopate. For this reason, therefore, inasmuch as they had obtained a perfect fore-knowledge of this, they appointed those [ministers] already mentioned, and afterwards gave instructions, that when these should fall asleep, other approved men should succeed them in their ministry. We are of opinion, therefore, that those appointed by them, or afterwards by other eminent men, with the consent of the whole Church, and who have blamelessly served the flock of Christ in a humble, peaceable, and disinterested spirit, and have for a long time possessed the good opinion of all, cannot be justly dismissed from the ministry. For our sin will not be small, if we eject from the episcopate those who have blamelessly and holily fulfilled its duties." (Clement of Rome, First Epistle to the Corinthians, 42 and 44)
This principle of an apostolic succession was explicitly articulated by other writers throughout the early Church, including writers such as St. Ignatius (Epistle to the Trallians, 3; Epistle to the Smyrneans, 8), who died in 107AD; the second-century Father St. Irenaeus (Against Heresies, 3, 3-4; 4:26; 4:33); and the third-century writers Tertullian (The Prescription Against the Heretics, 20-21, 32), Hippolytus (Apostolic Tradition, 1-2), and St. Cyprian of Carthage (On the Unity of the Church, 5 and 10; Epistle 33; Epistle 66; Epistle 69). In other words, not only is the concept of a hierarchy of apostolic origin, but since the time of the apostles there has also been an acknowledgement that not just anyone is part of this Church hierarchy, but that specific people are appointed as successors.
This hierarchy had the responsibility of teaching the faithful, setting a good example, doing Christian deeds to help build up the kingdom of God, facilitate the reception of the sacraments and cultivation of the virtues, and correct others who had gone astray in their doctrine, even if they were other leaders in the Church. (Tit. 3:10; 3 Jn. 9-10; 2 Tim. 4:5).
In essence, there was a visible hierarchy and there was suppose to be accountability within this hierarchy. Even Peter was accountable, as Paul well knew (cf Gal. 2:11-21). This is the reason that, even when someone was especially chosen outside of the regular procedures to be a leader, this appointment still had to be authorized by the Church. What happened after St. Paul's conversion is a profound and telling thing in this regard. St. Paul was given a special commission and taught "his gospel" directly from God, and yet he felt that he needed to make sure he was in doctrinal agreement and accepted by the apostles! (Gal. 1:11-12, 17-19; 2:1-10). Also, St. Paul tells us himself that he was healed and confirmed by someone in the Church, before he started preaching his gospel (Acts 9:17-20). Even a "direct line" (so to speak) with God did not give one the right to work apart from the appointed leadership.
It is no suprise, then, that more than once in Acts we see the hierarchy of the Church gathering to consider how to resolve certain issues. The Council as recorded in Acts 15 has already been mentioned. It is noteworthy that the Council sent their decision out expecting that their ruling would be followed and affirmed by everyone in the Church (cf Acts 15:28-30). They didn't put it up for a vote, as though each parish got to decide what they wanted to do. The Church (and specifically the hierarchy) had made a decision, and this decision was expected to be followed. And it was. When issues arose, and some clarification was necessary, another council was convened (as recorded in Acts 21).
And so it has continued throughout Church history, that the hierarchy which has its succession directly from the apostles, leads the Church year by year, gathers in council together to discuss and resolve various matters when necessary. Hierarchy and administrative church order, while admittedly remaining very loose on an empire-wide level for the first few centuries, was nonetheless visible, and people could not spontaneously rise up and declare themselves leaders; you had to be appointed by the rest of the Church leadership. And, this leadership was supposed to be "peer reviewed" by all the other hierarchs in the Church, so that actions were taken when someone went too far outside the bounds of orthodoxy or proper conduct.
It was a hierarchy of servanthood, collegiality, conciliarity. It is a phenomenon where the leader is called "first among equals": first because someone has to lead, if only to make sure that every bishop gets a say, and equals because (substantially or qualitatively) no one bishop is greater than another. The first among equals acts in a validatory manner: helping and confirming, rather than ruling and pronouncing. And when this happens, the hierarchy works as Christ our God intended.