As the previous post stated the word, "catholic" means universal. So, the catholic church is claiming to be the uinversal church.
When we add the adjective, Roman, we are referring to one expression of Christianity. Currently there are three major branches and at one point, there was a fourth: Catholicism, Orthodoxy, Protestantism and the Syrian church.
In terms of the beginning of the church, there is no one answer. So, let me give you a few different perspectives. First, some people will say that the chuch existed in the Old Testament, since they had faith in God. For example, they say that the assembly of God's people in the Old Testament equals the church. Second, some people will say that the church started with Jesus' ministry and his promises to build his church (Matthew 16). Third, others will say that the church came into existence through the works of the apostled who were commissioned by Jesus to establish his church (Ephesians 2:20).
Roman Catholicism is Catholicism that is closely aligned with Rome in theology and finds it allegiance with the Pope. To explain the differences among Eastern Rite Catholics and Roman Catholics, one must look back in history.
At the Nicea Council of 325 A.D. there was some disagreement over the divinity of the Son of God, Jesus Christ. For several decades this debate continued. Then, in subsequent councils there were other disagreements on theological teachings about the Holy Spirit. After more ecumenical councils, there became a division of the Eastern Churches with Rome. During this time, the Byzantine, Armenian, Syrian, Chaldean, Maronite, Coptic, and Greek Orthodox as well as the Russian Orthodox Churches parted from the Roman Catholic Church.
The word "catholic" with a small "c" just means something like "applying to everyone" or "universal." The Catholic church got that part of its name from the fact that it was the universal Christian church for a long time. The "Roman" part of the name comes from the fact that the popes have been the bishops of Rome and have almost always lived in Rome -- so the church is headquartered in Rome.
As far as the beginnings of the Church go, you have Matthew 16:18 where Jesus tells Peter he is the rock the church is built on -- this is seen as making him the first Pope.
You can also see the beginnings of the Church in the Acts of the Apostles and in all of the epistles of the New Testament -- these are documents about the way the church was as it started.