Importance of Papacy
The message sent to Saint Peter
"I will give you the keys to the kingdom of heaven whatever you bind on earth shall be bound in heaven"
This is an example of how Peter is very important to the papacy.
The Popes authority and universal jurisdiction over the church has had opponents. It's teaching has origins directly from Christ commissionings of Peter to be the visible representation of the church
Primacy of Peter
And I tell you that you are Peter and on this rock I will build my church and the gates of Hades will not prevail against it
Second and First Vatican Council
The second Vatican Council taught on the primacy of the Pope. They also began its teachings on the ministry of the bishops and then address the Pope.
The first Vatican Council defined papal infallibility. Which holds that the pope is preserved from any errors when teaching dogmatically about faith and morals.