The phrase "infallible pope" means that according to Catholic belief, the Pope is considered incapable of making mistakes or being wrong when he speaks on matters of faith and morals. Full definition
The former INFALLIBLE popes are rolling in their graves, no doubt. (religion.blogs.cnn.com)
One day he is a fallible cardinal and the next an infallible pope. (religion.blogs.cnn.com)
Both, ultimately, subscribed to the same vision of God as sovereign ruler rather than loving father, and it was this vision that supplied the rationale behind autocratic rule, whether that rule came in the form of an inerrant scripture, an infallible pope, or an all - encompassing state. (firstthings.com)