Pope Snowball's story begins with the original Pope, Saint Peter. On the night that Jesus was slated to die by Crucifixion, he selected one of his followers to be the new leader of his church. Knowing that one man could never be the voice of a faith due to how easily they could be corrupted, he instead selected his most trusted and pure disciple. That disciple was a rabbit named Peter, who became the first Pope. However, as the years passed, the rabbit Peter was replaced by a human Pope, and the Church hid the truth of the first Pope.
Peter went on to have children, all of whom were protected by the mysterious Hare Club For Men, with Snowball being the most recent of descendants. However, when Pope Benedict XVI and Bill Donohue teamed up to destroy the Hare Club and end Peter Rabbit's bloodline once and for all, it came to Stan and Kyle to protect Snowball. Stan, unfortunately, chose to save his father (who was a hostage in Vatican) over Snowball and gave the bunny over the Donohue, under the requirement that no one, not even the bunny would be hurt. They were, however, double-crossed (literally). Jesus however managed to arrive and save Snowball from being cooked into a cannibalistic human/rabbit stew (the Hare Club members where intended to be cooked into the stew along with Snowball).
After Donohue was killed by Jesus, Snowball officially replaced Benedict as Pope.
When asked by his Cardinals how to run the church, Snowball remains silent, just as Jesus intended.