I sometimes hear that the pope is infallible, and other times I’m told the church is infallible. I’m curious to know how infallibility is defined in Christianity, but more specifically in Catholicism. What does it mean to be infallible, what’s its scope, and who is infallible?. I find this fascinating because Muslims also believe in infallible beings. More so in Shia School of thought than the Sunni.
You are asking a flawed question because you have not defined any terms. I can only answer it by defining these terms.
Infallible, in Catholic thought, means unable to teach errors in issues of Faith and Morals. Period. That is as far as it goes.
The Church is infallible only when you mean “Chuch” as being the Bishops in union with Rome. In other words, the Body known as the Magisterium is incapable of teaching errors in matters of Faith and Morals when they are issuing teachings on these matters.
The pope is infallible when he is issuing teachings on matters of faith and morals from his official position as Pope (known as speaking ex cathedra).
Infallible does not mean unable to sin. It does not mean perfect. It is a very limited protection.
Now, the Eastern Orthodox do not have the same understanding of the Pope as the Western Catholic Church. To them, the bishops when teaching as a group (ecuminical councils or synods) are infallible.
None of us teach that any of these men are unable to sin.
The Church’s teaching on infallibility, its scope and its limits are defined in Chapter 4 of Session 4 of the First Vatican Council, available here.
The Pope isn’t infallible. He’s fallible. However, he exercises infallibility at certain times, e.g. in 1950 when the Pope dogmatically defined the Assumption of Mary (much like St. Paul exercised infallibility when he wrote the Epistle to the Romans).