It’s a terribly misinformed, but common anti-Catholic trope: “Catholics,” it is said, “have to believe everything the pope says.”
Catholics do not have to believe “everything the pope says.” But, on the other hand, the Church teaches in Lumen Gentium, Vatican II’s Dogmatic Constitution on the Church, “In matters of faith and morals, the bishops speak in the name of Christ and the faithful are to accept their teaching and adhere to it with a religious assent.” So while we don’t have to believe “everything the pope says,” when he teaches us as pope, Catholics are conscience bound, to accept his teaching and adhere to it with “a religious assent.”
What does that mean on a practical level as we hear or read papal sermons, encyclicals, exhortations, and other writings? What if we sincerely disagree with him? How do we sort it out?