I was recently at the Vatican Museum (i’ve been quite often) listening to the audio guide as I meandered along the fig leaf imbued statues, when I happened upon one of Pope Pius (not sure which). And the audio guide explained to me that the Pope is “Christs representative on Earth”.
I am having a hard time comprehending this. Could any Catholics out there explain it to me? I am planning on taking the train to Rome in the morning to see this man for myself.
I think the Catholic position is that Jesus said to the Apostle Peter, “I’m putting you in charge down here while I’m gone.”, and when Peter died, that authority passed down to somebody else, and so on, until you get to the current Pope.
IANARC but IIRC the designation is based on the biblical story of Jesus designating his apostle Peter as “the rock upon which I will build my church.” Nice little pun there from Jesus. Peter was the first pope. This site lists all the popes going back to Peter. Or I should say, it lists all the popes that the current RC heirarchy recognizes. It seems to gloss over those times when there were multiple people claiming the title (and I can’t seem to find Pope Joan anywhere!).
Also remember these words from Jesus to Peter: “Whatever you bind on Earth shall be bound in Heaven; whatever you loose on Earth shall be loosed in heaven.” This is taken to mean that Peter (and his successors) hold "the keys to the kingdom of Heaven). This is why the crossed keys are the symbol of Peter, and why popular myth (and a million "Pearly Gates jokes) have St. Peter at the gates of heaven).
The Catechism of the Catholic Church discusses the mission and organization of the church beginning in section 871 with the expression of the roles of the bishops and pope in sections 879 through 896. Those in the Orthodox and Anglican communions would probably agree with much (not all) that is said of the bishops, here, but would be quite firm in their opposition to the claims made for the pope in these passages. Those Christian communities that are congregational rather than episcopal in their organization, would probably reject much of what is said of the bishops, as well.
It wasn’t a pun, since “Peter” 's name was Simon. If the same thing was taking place now in an english-speaking country, Jesus would state that ** Otto ** is the rock upon which he will build his church, and from then on, Otto would be known as Mr Rock.
Of course, there are people arguing that the “petrus” thing comes from the bad translation in latin of a nickname, Simon KEPHAS, supposedly refering to him being bald. But that’s only an unprovable hypothesis. And extremely tangential to the OP.
If The Rock was one of the Twelve Apostles, does this mean that we need to add Austin 3:16 to the canon? 
Great answers thanks. It sounds like the structural organization of the Catholic church is very similar to that of the Mormons. If not tit for tat at least in general.
Does the Pope take on any role of prophetising? (i.e. the word of God comes through him) or is he just a figure head?
It is held that the Pope can be inspired by God, and that at such times (speaking ex cathedra – “from the throne”), his judgements are infallible. However, this is not the same as prophecy, and I believe that Catholics agree with the rabbis that the time of prophecy is past (though they may not agree on when that time ended!)
As others have said, Jesus singled Peter out to be his #2 man(“I call him…number two”). The Christian church later came to the decision that Peter was in essence the first Bishop of Rome (because he died in Rome?) and since the Bishop of Rome later became Pope, Peter posthumously becomes the first Pope in the eyes of the Roman Catholic Church, and thus, because Peter was Jesus’s spirtual sucessor and Peter was the first Pope, then all the popes afterwards were in essence spirtual sucessors to Peter and “God’s Representative on Earth”.
Though only the Catholic Church feels that way, apparently. The Eastern Orthodox Branch apparently feels that the Bishop of Rome(The Pope) has no more authority from God then the other Bishops. The Protestant churches see the Pope as the head of the Catholic Church but no greater.
You can probably imagine that Early Christians probably wished Jesus had left more instructions on just how he wanted the whole thing to be run before he left. One could easily argue if the trust Jesus put in Peter continued through the other members of the church, or died with Peter, something like “Peter, I trust you to carry on my work, but that does not give anyone else special authority”.
Jesus could be awfuly vague at times.
Or wonderfully vague, depending on your point of view, I guess.