Use the noun papacy when you're talking about the leader of the Roman Catholic Church and the particular authority he holds. The papacy supports reducing poverty, but it does not support women becoming priests.
The pope is the head of the Catholic Church in Rome, and his office or government is the papacy. You can use the word for official positions the church holds, or to talk about the history of a pope's term. For example, you might say, "The papacy of Pope Benedict XVI ended when he resigned." The root of papacy is the Latin papatia, "papal office," which comes from papa, or "pope."