This is the first in a series of "The Big Ten" doctrines that non-Catholics dispute. According to the Catholic Church, the Pope is the successor of St Peter and holds jurisdiction over the entire universal Church. The doctrine of the Universal Jurisdiction of the Papacy is defined in the Catechism of the Catholic Church in this way:
882 The Pope, Bishop of Rome and Peter's successor, "is the perpetual and visible source and foundation of the unity both of the bishops and of the whole company of the faithful." "For the Roman Pontiff, by reason of his office as Vicar of Christ, and as pastor of the entire Church has full, supreme, and universal power over the whole Church, a power which he can always exercise unhindered." (Lumen Gentium 22)
Part 1: Luke Chapter 12
Now this seems a bit much to both Protestants and Orthodox Christians. Did Christ give one man so much power? St Peter himself seemed to have questions about this:
Luke Chapter 12
 Peter said, "Lord, are you telling this parable for us or for all?"
 And the Lord said, "Who then is the faithful and wise steward, whom his master will set over his household, to give them their portion of food at the proper time?
 Blessed is that servant whom his master when he comes will find so doing.
 Truly, I say to you, he will set him over all his possessions.
Christ says that the Master (i.e. Himself) will set a steward "over his household" and "set him over all his possessions."
Okay, so Christ has said it, we know it is true. Now that means Christ has a steward "over all His possessions." He didn't set an Ecumenical Council "over all His possessions." No, this universal jurisdiction is given to one single "steward." If you look at Luke 12:41, you will see that these words are addressed to Peter.
Non-Catholic Christians need to ask themselves, "If I believe the Bible, I must believe that Christ has set a steward over His Church until He comes again in glory. Who is this steward who governs for Christ and why am I not under him?"
Part 2: Matthew Chapter 16
 Simon Peter replied, "You are the Christ, the Son of the living God."
 And Jesus answered him, "Blessed are you, Simon Bar-Jona! For flesh and blood has not revealed this to you, but my Father who is in heaven.
 And I tell you, you are Peter, and on this rock I will build my church, and the powers of death shall not prevail against it.
 I will give you the keys of the kingdom of heaven, and whatever you bind on earth shall be bound in heaven, and whatever you loose on earth shall be loosed in heaven."
Here we see that St Peter is not just given preeminence as "first among equals." Rather, Christ explicitly gives one man the keys of the kingdom. "Whatever you (here the Greek is singular - denoting one man) bind on earth shall be bound in heaven." Thus, Peter had authority over all the earth. That is universal jurisdiction.
If if one should deny that the Petrine Ministry was not passed down through the ages to the Popes, still one must grant that during his life, St Peter did have Universal Jurisdiction to "bind on earth." This power is not relegated to "the West" but to "earth."