The Bishop of Rome is the Pope, not vice-versa.
When the Cardinals enter into conclave sometime in the next few weeks, it is not so much to elect a Pope as it is to choose a bishop for the diocese of Rome. That man will be Pope in virtue of his being the bishop of Rome.
The conclave itself hints at the way bishops were chosen in the first centuries of the Church. In some areas, all the faithful had a voice (that is basically how St. Ambrose was chosen as bishop of Milan in 374). In other local Churches, the clergy came together as a “college” or congress to pick the one who would be chief shepherd among them. In a city the size of Rome, this college was composed of the leaders of the original “parishes” of Rome—which you can still visit today (though their ruins are ten feet below the ground level of modern Rome). Even now, each Cardinal is honorary pastor of a Roman parish, called his “titular” Church, so that symbolically, at least, the bishop is still chosen by a group of Roman clergy.
But why Rome? Wasn't Jerusalem the center of Christianity in biblical times? Why isn't the bishop of Jerusalem the Pope?
This hearkens back to Jesus choice of not a city, but a particular person to be the “rock” on which he would build his Church. That person was Peter.
If Jerusalem had remained Peter's center of ministry, then today's bishop of Jerusalem might indeed have been Pope (most likely of a tiny, strongly Jewish Christian community—if anyone survived the destruction of the city by Rome in 70 AD).
In God's providence, Peter realized early on that he had a universal ministry: he himself “gave orders” that the devout Gentile Cornelius be baptized (Acts 10: 34-38) without first becoming a Jew through circumcision. With an increasing number of Gentile converts in the vast city of Antioch (modern day Antakya, Turkey), Peter moved there. (The Maronite Patriarch of Antioch boasts of being “the successor of St. Peter in the See of Antioch.”)
But Peter did not stay in Antioch: like Paul, he was drawn to the capital city, to Rome. And it was there that he died, by order of Nero, crucified in the “circus” (racetrack) by the Collis Vaticanus (Vatican Hill) across the Tiber from the city center. From then on, every chief shepherd of the Church of Rome could only ever be the successor of St. Peter.
It is this connection to St. Peter that designates the bishop of Rome not just the shepherd of one diocese among many but the visible center of unity for all the churches of the world. In other words, the Pope.
The North American College (seminary) has an interesting list of the ancient parishes ("station churches") of Rome, along with a Google map (with photographs of some of the churches) and other features.
This is #2 in a series of 7 Things Every Catholic Should Know about the Papacy.
Here is the #1 Thing Every Catholic Should Know
For a History of the Popes by a reliable Catholic scholar, try the one by John O'Malley, SJ.