The expression comes from a legend which has been used to affirm the Catholic belief that the Pope must reside in Rome and Rome alone.
According to the legend, the Apostle Peter (whom the Catholic Church counts as the first Pope, or, more precisely, the Catholic Church considers the Pope to be Peter's successor as the holder of the keys) decided to leave Rome when the Emperor started the first persecution of Christians.
As he was walking, already a couple miles out of Rome, he met Jesus walking in the opposite direction.
Puzzled, Peter asked: "Quo vadis, Domine?" ("Where are you going, Lord?")
To that Jesus replied, "I'm going to Rome to get crucified again."
Peter got the point, and returned to Rome. He was later crucified himself and buried at the Vatican, which was one of the hills of Rome.
There is a Quo Vadis chapel on the outskirts of Rome (on Via Appia, if I remember correctly) where this legend was supposed to have happened.
Also, Quo Vadis is a novel by Henryk Sienkiewicz. The movie was probably based on the novel.