To answer the question of origins, Jesus appointed 12 apostles and from these men, the Vatican, the Pope, all the bishops and even the priests not only come from, but derive their authority from.
Peter was made the head of the Christian Church, able to confirm his brethern, as we see in Matthew 16:18, he is given the Keys to the Kingdom of Heaven and his role in the faith is the foundation of the Church.
At Pentecost, the Holy Spirit came upon the apostles and their mission began. First they spread the Gospel to the local jews, but when the jews did not accept Christ as a whole, they began preaching to the Gentiles.
Local Churches began sprouting up around the world, as the apostles and disciples began preaching. As we see recorded in the bible, the regional Churches had bishops in charge of them. Bishops are men appointed by apostles and carry the authority that Christ gave to the first twelve.
During this time, the Gospel was spread and it was later recorded in the four Gospels we know today. Some of the stories of the apostles, narrated by Luke, became the Acts (of the apostles). Paul, who went around the world preaching, wrote numerous letters to the various Churches and these made up the bulk to the New Testament. Finally, other letters were written directed at all Churches, which became known as the "Catholic letters", as they were meant for the univeral Christian audience. Finally, John wrote Revelation while on Patmos and the scriptures that make up our bible today were written.
During that time, Peter (along with Paul) and his brother Andrew went to the two capitals of the Roman Empire- Rome and Constantinople, respectively. There the men established Apostolic Seats = Sedes = Sees. Peter had already gone and established one in Antioch, while James is credited with the first, at Jerusalem. Mark also established a See at Alexandria.
Peter and Paul were martyred in Rome and their successors have stayed there for the last 2000 years. The Vatican is what we call Peter's See, because Peter's successors later built their buildings on Vatican hill, where Peter was buried. Just as Benedict XVI replaced John Paul II, this process has been going on since Peter founded it (who was replaced by Linus).
Each priest in the Catholic, Orthodox and Coptic Churches was appointed by a Bishop and each Bishop was appointed by an earlier Bishop, which goes all the way to the apostles. This is called "apostolic succession".
What is taught in these three Churches is Holy Tradition, teachings given to us by the apostles. Most of what they taught was recorded in the bible in some way. Whenever controversy came up in the Church, Ecumenical Councils were called to deal with it. The fives Sees founded by the apostles would send bishops, who, inspired by the Holy Spirit, would cover the issues. What was produced is called "doctrine".
These Councils latered ruled on what texts would be considered inspired. This canon of texts declared to be inspired are what make up our bible today.
In the 5th Century, the Churches of North Africa departed from the rest of Christianity and became known as the Coptic church.
In the 11th Century, the Churches of the east departed from the Churches of the west and became known as the Orthodox church. The west Churches retained the name "Catholic Church", which had been used to refer to the Churches that make up Christianity since the 1st Century.
In the 16h Century, King Henry VIII of England had his nation pull out of the Catholic Church and became the head of the parishes there, forming the Anglican Church.
Calvin, Martin Luther and other men either looking to reform the Catholic Church or begin their own churches also started tearing away and developing new doctrines. Doctrines based not on what the original apostoles taught, but their interpretations of the bible.