We can use the Immaculate Conception and the Assumption of Mary into Heaven as examples. These were discussions that started "from the pews" as the saying goes, meaning it was being discussed by parishioners. This discussion was taken up by the dioceses, then the archdioceses, then by the Cardinals at the Vatican. There was no real agreement all up the line, and after hundreds of years it came before the Pope. Here is where the Pope made an infallible decision, meaning that this decision was final and could not be changed by any future popes. If I recall correctly, these are the only two infallible decisions made by a pope. (Yes, in all of Church history, the infallibility of a pope has been exercised only two times.)
Let's take another for instance where some Christians no longer wanted to observe all seven Sacraments. They left the Catholic faith and started a new denomination. Even so, Christians who separate themselves from the Catholic faith are still considered members of the Body of Christ even though they are not in full communion with the Catholic faith.