Catholics compiled the Bible by deciding which of the Jewish holy scriptures they believed the Holy Spirit was directing them to include making up what Christians call the Old Testament. They also discerned which Christian writings since the time of Christ should be included to create the New Testament. Old Testament + New Testament = Bible. This compiling of the bible happened centuries after Christ's Ascension, Jesus obviously did not walk around handing out Bibles (especially as the printing press was not invented for many more centuries and the Jewish people kept their holy scriptures on hand written scrolls). Despite not having a bible for centuries (although of course holy writings would have been used) the church grew through the teaching and evangelising of the Apostles. Therefore it was oral teaching and tradition together with scripture that formed the church. Just like the Jewish people the Christians had a history of interpreting the scriptures through tradition and oral teaching. The Christian community therefore had three sources of authority: Scripture, Sacred Tradition and Teaching Authority (from the Apostles). The teaching authority is called the Magisterium which is vested in the Pope and bishops who are in communion with him. Sacred Scripture (the Bible) and Tradition "make up a single sacred deposit of the Word of God, which is entrusted to the Church".
The office of the Pope developed from St. Peter the Apostle which we see in scripture
"Simon said "You are Christ, the Son of the living God". And Jesus answering said to him "Blessed are you, Simon-Bar-Jona: because flesh and blood has not revealed this to you, but my Father who is in heaven. And I say to you: You are Peter [Aramaic Kipha, a rock], and upon this rock [Kipha] I will build my church and the gates of hell shall not prevail against it. And I will give to you the keys of the kingdom of heaven. And whatsoever you shall bind upon earth, it shall be bound also in heaven: and whatsoever you shall loose on earth shall be loosed also in heaven."
Matthew 16:13-20; Mark 8:27-30; Luke 9 :18:21
The line of Popes is unbroken from our current Pope Francis to St.Peter and St.Peter's church in the Vatican, probably the most famous church in the world, is built over the tomb of St. Peter which can be visited in the crypt below. The vast majority of Catholic doctrine is based on scripture anything else comes from Sacred Tradition. The Bible was unchanged for 1500 years (so 3/4 of the church's history) until the Reformation when Luther and other Protestants removed a number of books which had always been considered part of scripture by all Christians. Catholics call these books Deuterocanonical and Protestants Apocrypha. The books Luther removed contain the scriptural teachings on Purgatory and asking for the prayers of those already in heaven (the saints).
The Catholic church (there was only one unified church at the time no Orthodox or Protestants) compiled the bible and therefore believes she has the authority to interpret scripture and what it means. All Catholics can of course read their Bibles and they hear scripture read from the Old and New Testament and from the Gospels at every Mass but in case of disputes the church uses her teaching authority of Tradition and the Magisterium to state clearly what we believe it says and means. This is how the teaching of the Trinity developed as it is not taught clearly as such in the scriptures.
The big difference between the Catholics and Protestants is Authority. Who has the authority to say what the Apostles taught and what scripture says? The bible ALONE does not hold up as we did not have the bible for many generations. If you read all the history of the early church (even before the bible was compiled we have written records) you discover early Christians believed in the same dogmas that modern Catholics do.
Without a teaching authority anything goes and if people disagree they can just walk down the street and open a new church. As a result you have over 30,000 protestant groups all aiming to teach from scripture. The church says it is One, Holy, Catholic (universal) and Apostolic.
The Canon or official list of Scripture was only compiled at the end of the 4th century at Hippo in 393, Carthage in 397 and where was it sent for confirmation? To Rome in 419.