The doctrine of the pope comes straight from the New Testament. You will not find the word pope in the New Testament, because it is an affectionate term applied to the office some time later. It comes from the Greek word πάππας (pappas), which means papa, and is not part of the official title of this office. The official title for the pope is this: Bishop of Rome, Vicar of Jesus Christ, Successor of the Prince of the Apostles, Supreme Pontiff of the Universal Church, Primate of Italy, Archbishop and Metropolitan of the Province of Rome, Sovereign of the State of Vatican City, Servant of the Servants of God. It is no wonder why many simply refer to this office as the pope. The most important part of the title is the last part, Servant of the Servants of God. This last part of the title illustrates not only the popes role, but the role of the bishops and priests as servants to the people of God.
The office of pope was established by Christ when he said to Peter, the first pope, "Blessed are you, Simon son of Jonah. For flesh and blood has not revealed this to you, but my heavenly Father. And so I say to you, you are Peter, and upon this rock I will build my church, and the gates of the netherworld shall not prevail against it. I will give you the keys to the kingdom of heaven. Whatever you bind on earth shall be bound in heaven; and whatever you loose on earth shall be loosed in heaven." (Matthew 16:17-19) The power given to this office, and the promise that it will not fall, is very clear.
The responsibilities of the pope are also bestowed by Christ. "When they had finished breakfast, Jesus said to Simon Peter, 'Simon, son of John, do you love me more than these?' He said to him, 'Yes, Lord, you know that I love you.' He said to him, 'Feed my lambs.' He then said to him a second time, 'Simon, son of John, do you love me?' He said to him, 'Yes, Lord, you know that I love you.' He said to him, 'Tend my sheep.' He said to him the third time, 'Simon, son of John, do you love me?' Peter was distressed that he had said to him a third time, 'Do you love me?' and he said to him, 'Lord, you know everything; you know that I love you.' Jesus said to him, 'Feed my sheep.'" (John 21:15-17) The first responsibility given to the pope is to love Christ; to love him "more than these." The second responsibility of the pope is to feed and tend Christ's Church. Christ is the shepherd, and the members of his Church are the sheep. The pope is not the shepherd of the sheep, but the shepherd's servant. The shepherd could not remain physically present with his sheep, so he entrusted his sheep to his loving servant, who will love and care for them, until the shepherd returns.
The pope is infallible in everything he does and says
The man that holds the office of the pope is just a fallible and anyone else, it is the office of the pope that is infallible. This means that when the pope, in communion with the Church, delivers objective, definitive teachings regarding faith and morals, they are without error. This is accomplished by the work of the Holy Spirit to fulfill Christ's words, "the gates of the netherworld shall not prevail against it (the Church)." (Matthew 16:18)
The pope is the head of the Catholic Church and not Christ
As the explanation of the pope's responsibilities are described above, Christ is the Head of the Church. In Christ's physical absence, he leaves the pope, to be the physically present head of the Church. This is where a portion of the pope's title comes from, "Vicar of Jesus Christ".
Because the pope is the Vicar of Christ, he stands between the people and Christ
In no way does the Vicar of Jesus Christ stand between Christ and his Church. The Vicar of Jesus Christ, at Christ's command, tends and feeds Christ's Church. However, it is Christ's Church, and Christ has an intimate relationship with each member of his Church.