The current Roman Catholic Church teaches the doctrine of Papal infallibility, which states that the bishop or Pope of the Roman catholic church should be the head of the entire church because he and he alone inherited the authority of St. Peter.

Scriptures
In Matthew 16:18 after Peter says Jesus is the Christ, to prove "on this rock I will build my church."
John 20:23 states what ever sins you forgive will be forgiven but the sins you do not forgive will not be forgiven.
The catholic position is that these verses refer to Peter.
Protestants argue that these two verses refer to the 12 and Christianity in general.
but in Matthew 18:18,19  clearly states that if any two or more of the 12 agree on anything in the lord's name it will be done for them, what ever they bind will be bound and what ever they loose will be loosed.
Matthew 19:28 the 12 would rule on 12 thrones judging the 12 tribes.
In Galatians 2 we see Paul correcting Peter for being to legalistic. The fact that there were anti-popes proves the pope is fallible. 
Acts 15 the Jerusalem council shows the body of Christ coming together to find the answers not simply appealing to Pope Peter.

The Roman Catholic Church must go back to a biblical standard otherwise they cause the church to be divided. As Clement of Alexandria says in Stromata book 2.19 "Whoever shall attempt to do aught with presumption, provokes God."

History
Irenaeus ~180 AD
Irenaeus in his Against Heresies 3.3 states that every church should agree with the church of Rome because it was founded by both Peter and Paul. This looks like papal infallibility but he explains this comment in 3.2. On account of Rome being the capital of the empire, the faithful from all parts, representing every Church, are obliged to resort to Rome, and there to come together; so that the apostolic tradition of "the gathering together out of all the Churches" will be preserved. (Later some Latin texts will give the impression of coming to hear rather than to debate) 

In Eusebius E.H. 5:20 Irenaeus corrects several schism's of Roman bishops, Blastus, Florinus, ect. Pope Victor is said to have blessed the false prophet Montanus, until Irenaeus brought to light his Heresy. At that time Victor excommunicates Montanus and his followers from Rome.

The council of Chalcedon 451 AD states Rome was given Primacy only because it was the capital of the empire, and is equal in privileges to Constantinople and the others.

Cyprian ~ 250 AD
When reading Cyprian's epistles on the "Baptism of Heretics" several things come to light:

1. In Epistles 16,29, and 30 Cyprian is addressed as "Cypriano Papae" or Pope Cyprian. So the word Pope is being used to the head of a group of churches (Greece, Italy, ect. Cyprian is Pope of Carthage and Stephen is Pope of Rome.
2. We see that Pope of Rome is equal to all the other bishops, and is called a colleague of Cyprian in epistles 3,17,31,51
3. In epistle 47 the Pope of Rome refers to the other bishops as "co-bishops"
4. The Keys to the kingdom and the power of binding and loosing refer to all bishops equally and decisions should be made with the consent of all the bishops in epistles 25,26

• 40 - Cyprian rebukes Rome for ordaining Novatian (unlawful ordination and in opposition to the Catholic Church) and will not recognize or be in communion with any who are a part of his faction.
• 50 - Cyprian instructs Rome on the unity of the church, "some are always assuming on themselves more dominion" with a "sacrilegious presumption" and a "proud obstinacy" and thereby "perish from the church" see also Cyprian's "Unity of the Church"
• 51-  After the persecution all the remaining Carthaginian bishops got together to debate what to do with the lapsed, both side appealing to scripture and the only authority. Then notified Rome of their decision. Rome had a similar council and came to the same conclusion. 
• 66 - Cyprian warns Stephen of Rome to excommunicate Novatian and his party. 
• 67 - Stephen was deceived by Basilides and Novatian. Cyprian states the prophecies of the Antichrist fore told of heresies rising up in the church at the end of the world.
• 70 - His colleague Stephen (Pope of Rome) is presumptuous.
• 73 - Cyprian wrote to bishop Stephen of Rome because he was in "error" by forbidding anyone coming from a heresy to be baptized in the church. Judging the baptism of heretics to be lawful. Stephen may have considered heretics to be only non-Trinitarians and therefore not consider Novatian a heretic. Stephen said that since the heretics do not rebaptize a convert who came to them from us, we should not rebaptize one of them who converts to the true church. 
• 74 - Firmilian bishop of Caesarea writes to Cyprian stating Stephan bishop of Rome was unkind and compares him to to Judas who betrayed Christ. He is thankful that Cyprian settled the matter one and for all. Firmilian says not only was Stephen wrong about baptism but also "They who are at Rome do not observe those things in all cases which are handed down from the beginning, and vainly pretend the authority of the apostles" There are small differences in each church due to culture and language but no break in unity. "The folly of Stephen" is that he "boasts of the place of his episcopate, and contends that he holds the succession from Peter." He does not understand that "the truth of the Christian Rock is overshadowed, and in some measure abolished, by him when he betrays and deserts unity this way". Stephen says baptism in the name of Jesus alone is OK, making heretic baptisms acceptable. Stephen in so doing “cut himself off from the unity of the church” and can be considered “apostate.” Stephen at the last called Cyprian a “false Christ, false apostle, and a deceitful worker”
• 75 - The schismatic tries to "establish a throne for themselves and assume primacy and the right of baptism" 

Cyprian in his treatise on the Unity of the Church, defines the doctrine of apostolic unity to be the equality of all Bishops.

• Unity of the Church 4 - Peter is equal to the other apostles in the authority of binding and loosing, and forgiving sins. Whosoever does not hold to this teaching of unity "strives against and resists the church" and does not hold to "true faith"
• Unity of the Church 10 - Those who "appoint themselves as prelates with out any law of ordination" are the chaff separated from the wheat.

• Origen 230AD
• Commentary on Matthew 12.14 and 13.31
  The power of binding and loosing of one who has the keys to the kingdom is not that he has authority in himself to make new laws about sin. But that he has the answers (doctrinal truth) about Jesus and the church. The power in teaching total truth is that it instantly sets free those who listen and obey, from the sin and its consequences on their life, or it completely binds them to their sin if they refuse to listen, making the consequences that much more severe. But if a person has incorrect doctrine then those who listen to them will be trapped deeper in sin. These no longer have the power of binding an loosing. One can't free with the truth if one does not know the truth.
  So if some one speaks rightly we must listen to them but if not, we must not listen to them lest we be bound unnecessarily.
• Commentary on Matthew 12.11 - The promise of the key to the kingdom were given not to Peter alone but to all the disciples (the church).

The following Early Church fathers stated that the "rock" Jesus will build his church on was not Peter but the statement of faith that Jesus is Messiah.

• St. Hilary, bishop of Poitiers in his second book on the trinity
• St. Chrysostom in his 53rd Homily on St. Matthew
• St. Augustine