A traditional Latin Catholic dictum which translates Rome spoken, case closed.
The dictum may be prone to misinterpretation as if the various Church offices in Rome had the final say, so I thought I'd elaborate a bit.
There are many Church offices in Rome. Just as the various departments in the White House are created by the President of the United States and function vicariously drawing from his jurisdiction, these offices are created by the Pope and function under his authority.
It may be appropriate to ask at this point just who the Pope is and where his power derives from.
First and foremost, he is the Bishop of Rome. Secondly, he is the head of the Catholic Church. Please note that other dignitaries, especially in the Eastern Orthodox Christianity are called Pope. I am in no way trying to argue either for or against their use of the title. I am simply describing the Bishop of Rome in this write-up. That is why, for the purposes of this write-up only, I shall use the term Pope as identifying the Bishop of Rome. Also, I am simply presenting the hierarchy of the Church rule as understood by the Catholic Church. I am not arguing whether it should or should not be, nor whether it is good or evil, or anything like that.
According to Christian Gospels, Jesus gave the keys to the Kingdom to the Apostle Peter. This the Catholic Church understands to mean that Jesus made Peter his vicar, and gave him the authority to speak in his name, thus making him the final arbiter in any dispute in the matters of faith and morals (and only in those matters).
Subsequently, Peter went to Rome. According to the Quo vadis legend, Peter wanted to leave Rome during the time of persecution, but was told by Jesus to go back. This is the basis of the idea that Rome and Peter are inseparable.
After Peter's death (he was crucified in an upside-down position), the next Bishop of Rome became his successor. It is the Catholic belief that the Bishop of Rome not only succeeds Peter as the head of the Church in Rome, but also holds the same power of keys to the Kingdom, i.e., he, too, is the final arbiter in the matters of faith and morals.
Whenever, in early Christianity there was a conflict or argument between two bishops, they would ask the Bishop of Rome to decide for them. Whatever his decision, it was final and immutable. Rome spoken, case closed. He was (and is) seen as the Vicar of Christ, hence whatever he decided was the same as if Christ himself had decided.
It was for several centuries that the Pope only made a decision when asked by other bishops. He was (and still is) considered first among equal, or also servus servorum Dei (servant of God's servants). I'm sorry I no longer remember which Pope was the first to make a decision without being asked. He saw some problem that he thought had to be addressed by him. So he made a decree motu proprio (of his own motion) for the first time. This set a precedent followed to these days: The Popes throghout history had made many decrees motu proprio (which is now the formal technical term for such decrees).
To this day, the Pope is technically first and foremost the Bishop of Rome and only secondarily the head of the Catholic Church. The word Catholic actually means Universal. According to Catholic beliefs, the Pope is the head of all Christianity and has the God-given (literally) right to make decisions concerning any Christian. Naturally, he usually does not make such decisions because it is perceived as creating an unnecessary burden. But, for example, for a long time the Canon Law rules about marriage were intended binding to all Christians, so the marriages of Protestants weve viewed as invalid by the Catholic Church. The law was then changed explicitly to bind only Catholics.
Now, because the Pope has the double function, he also has two separate sets of offices. One is in the Lateran, which is the Chancery of Rome. There he functions as the local Bishop of Rome, the head of the Diocese of Rome.
The other, and more famous (as well as much bigger), set of offices is in the Vatican (built there because the Apostle Peter was buried there). There he functions as the head of the Universal Church. These offices are generally called congregations, just as the White House offices are called departments. It is the various congregations that make binding decisions for the entire Catholic Church.
Nevertheless, the dictum Roma locuta, causa finita still aplies to the Pope personally. For example, the decisions of the Sacra Rota Romana can be appealed. But Pope's decisions are final and without appeal.
To this day, the Pope still claims his authority only in the matters of faith and morals. Thus, he cannot decide to change the value of pi, for example, nor does he claim he can.