Introduction
The early church had a church polity that had evolved through several centuries from being apostolic to presbytery culminating to predominantly being mono-episcopate.
FIRST CENTURY (1 to 100 A.D.)
Apostolic
The most ancient church leaders were the apostles. The preeminent of these was Peter, having been the first one to confess the true faith that Jesus was the Christ, the Son of God (Matt.16:18). By his confession of the faith, Peter was called Cephas (''rock'') by the Lord Jesus. However, in his role of establishing the church , the Lord Jesus equally shared this to the other apostles (cf. Matthew 16:19; 18:18). Paul was clear and emphatic that the foundation of the early church were the apostles and the prophets, Jesus Christ himself being the chief cornerstone (Eph. 2:20).
Only Catholicism teaches that the Roman bishop should be the new 'rock' (Peter) to lead the church of the next generations.
It is noteworthy that the Roman bishop did claim this to himself whilst the church disagreed. We have Cyprian (200-258 A.D.) , bishop of Carthage, in the 3rd century very explicit that he was against papacy.
7th Council of Carthage: http://www.ccel.org... remains, that upon this
same matter each of us should bring forward what we think, judging no
man, nor rejecting any one from the right of communion, if he should
think differently from us. For neither does any of us set himself
up as a bishop of bishops, 4675 nor by tyrannical terror does
any compel his colleague to the necessity of obedience; since
every bishop, according to the allowance of his liberty and power, has
his own proper right of judgment, and can no more be judged by another
than he himself can judge another. 4676
Cyrian believed that the the other apostles were the same as Peter in regards to both honour and power:
Treatise 1, Chapter 4 http://www.ccel.org...... He arranged by His
authority the origin of that unity, as beginning from one. Assuredly
the rest of the apostles were also the same as was Peter, endowed
with a like partnership both of honour and power; but the beginning
proceeds from unity.3110 [Which one Church, also, the Holy Spirit in
the Song of Songs1 designated in the person of our Lord, and says,
“My dove, my spotless one, is but one. She is the only one of her
mother, elect of her that bare her.”]
However, in his Treatise I: On the Unity of the Church , Cyprian did believe in the primacy of Peter referring to the chair of Peter (cathedra Petri) as the source of that oneness or unity by which the church existed. Take note that the primacy of Peter (being held by Orthodox and some Protestants) is very different from papacy.
Presbytery
The New Testament had the original polity of presbytery (''πρεσβυτερίου' - I Tim. 4:14). This church government is the ruling of bishops over a local church. In the earliest days of the church, bishop and elder are synonymous. The elders were functioning as bishops (''overseers'').
In Acts 20, Paul's ministry was going to finish soon when he gave a final word of exhortation to the church in Ephesus. He began in verse 17 by addressing the πρεσβύτερος (elders) of the church, yet a few verses later in the passage, Paul referred to these same men in verse 28 as ἐπίσκοπος, or overseers. He described the same group one time as presbuteros and a second time as episkopos. Paul considered these two words to be synonyms for church leaders.
17 Now from Miletus he sent to Ephesus and called the elders of the
church to come to him. 18 And when they came to him, he said to
them: “You yourselves know how I lived among you the whole time from
the first day that I set foot in Asia, 19 serving the Lord with all
humility and with tears and with trials that happened to me through
the plots of the Jews; 20 how I did not shrink from declaring to you
anything that was profitable, and teaching you in public and from
house to house, 21 testifying both to Jews and to Greeks of repentance
toward God and of faith in our Lord Jesus Christ. 22 And now,
behold, I am going to Jerusalem, constrained by the Spirit, not
knowing what will happen to me there, 23 except that the Holy Spirit
testifies to me in every city that imprisonment and afflictions await
me. 24 But I do not account my life of any value nor as precious to
myself, if only I may finish my course and the ministry that I
received from the Lord Jesus, to testify to the gospel of the grace of
God. 25 And now, behold, I know that none of you among whom I have
gone about proclaiming the kingdom will see my face again. 26
Therefore I testify to you this day that I am innocent of the blood of
all, 27 for I did not shrink from declaring to you the whole counsel
of God. 28 Pay careful attention to yourselves and to all the
flock, in which the Holy Spirit has made you overseers, to care
for the church of God,[e] which he obtained with his own blood. (Acts
20:17-28 ESV)
We also have the Didache, a 1st century Christian document (ca. 65-80 A.D.) affirming this New Testament polity of elders/bishops.
"Appoint, therefore, for yourselves, bishops and deacons worthy of
the Lord, men meek, and not lovers of money, and truthful and
proved; for they also render to you the service of prophets and
teachers." (Didache Chapter XV. -Bishops and Deacons; Christian
Reproof.)
Monoepiscopacy
As early as the first century, churches had a polity that evolved rapidly from the presbytery to monoepiscopacy. In the New Testament, we had seen that such polity existed already in some churches notably of John's.
I wrote somewhat unto the church: but Diotrephes, who loveth to have
the preeminence among them, receiveth us not. (3 John 1:9 ASV)
We see in this passage that Diotrephes abuses his authority as leader in his local church. He loved more his position of being first rather than loving the family God has given him.
SECOND CENTURY TO THIRD CENTURY (101-300 A.D.)
Diocesan Monoepiscopacy
The church polity has evolved to having a bishop ruling not just his local church but other local churches as well. An instance is the bishop of Rome in the second century.
The first bishop to exercise jurisdictional authority was Victor I of Rome (r. ca 180 A.D.).
''Victor’s actions were more characteristic of a pope than those of
preceding bishops of Rome.'' (Source)
"Councils and Dioceses. We first hear of the bishops from local
congregations in the same province meeting for consultation in the
second half of the second century in connection with the problem posed
by Montanism. These meetings were first held in connection with common
problems but became regular gatherings. As churches in the large
cities grew, presbyters (elders) were assigned to particular
assemblies within the city. Earlier all congregational functions had
been under the supervision of the bishop. Increasingly these had to be
assigned to the presbyters. Christians living in outlying areas
would not have their own bishop but would look to the city church for
leadership. Thus the basis was laid for the bishop to preside over
several assemblies, although in theory all remained one church. The
territory presided over by a bishop is now called a diocese."
(Everett Ferguson, Early Christians Speak, p 16)
FOURTH CENTURY (301-400 A.D.)
The Metropolitan bishops
The church polity has evolved further and now we have a bishop ruling not just local churches but every local church in a city (metropolitan bishop).
The bishop of the capital city of a province (the metropolis) or of
another principal city began in the third century to assume a leading
position among the bishops of the province. He presided at the
provincial councils, gave his approval for the ordination of bishops,
and often was the principal ordainer himself when a new bishop was
installed. The metropolitan bishops of the ancient church were
forerunners of the medieval archbishops in the western church. The
idea of councils of bishops led to the calling of ecumenical councils
(world wide) to represent the whole church, the first of which was
summoned by the emperor Constantine for Nicaea in 325. The council
at Nicaea recognized the churches at Rome, Alexandria, and Antioch as
having jurisdictional authority extending beyond the usual provincial
limits. This was the germ of the patriarchal system. Eventually five
patriarchs were recognized: the bishops of the above three churches
and Constantinople and Jerusalem. When the division between the Greek
(or Eastern) churches and the Latin (or Western) churches occurred,
the Greek Church continued to hold to the patriarchal theory of church
organization, whereas the Latin Church had recognized the bishop of
Rome as the single pope thus giving it a monarchial organization."
(Everett Ferguson, Early Christians Speak, p 16)
FIFTH CENTURY (401-500 A.D.)
The Patriarchs
In the fifth century, the church polity had evolved even further. Now we have a bishop ruling over a geographical site (i.e. nation). Such is called the Patriarch. In this century, there are five major patriarchs. One from the West. Four from the East: Rome, Antioch, Alexandria, Constantinople and Jerusalem.
"The first ecumenical council of Nice, in 325, as yet knew nothing of
five patriarchs that would be fully established in 451 AD, but only the three metropolitans (Antioch, Alexandria, and Rome),
confirming them in their traditional rights." (Philip Schaff, History
of the Christian Church, book 3, chapter 5)
Discussion
The church polity from fourth century to fifth century was the same. Only Catholicism claimed that the traditional primacy of the bishop of Rome was really a Papacy or jurisdiction of the bishop of Rome over all churches both in the east and the west.
The East–West Schism of the 11th century was a product of centuries of differences (whether when to celebrate Easter, priests should be celibate, rules on fasting and of authority) between the Latin and Greek speaking churches. The bishop of Rome and the bishop of Constantinople had excommunicated each other due to the fact that the former dogmatically asserted his papacy over all bishops while the latter refused to acknowledge his papal authority.
Conclusion
The church structure from 300 A.D. to 500 A.D. was the same in both the East and the West. They were having monoepiscopacy that had evolved from metropolitan to patriarchy. Today the Orthodox remained the Patriarchy while in Roman Catholicism it was the Papacy.