There were originally, I believe, 5 Patriarchates (though I may be wrong). These were Rome, Antioch (in Greece), Jerusalem, Constantinople (now Istanbul in Turkey) and Alexandria (not sure about the last one, though). The Bishop of Rome (Pope--which is really an honourary title given to a number of Patriarchs in different denominations meaning "father") is the Patriarch of the Catholic Church's Latin Rite. He is also the Universal earthly head (beneath Christ and His representative) in Catholic thought.
The EO believe that from the beginning all 5 were equal, but Rome had primacy of honour as it's Bishop was "the first among equals"--that primacy being the only difference between the Patriarch in Rome and the other 4. Catholics believe that from the beginning the Pope (Bishop of Rome and Patriarch) was supreme to the others, because we believe he was also chosen to "Feed My sheep" as Christ says and was made "the rock on which I will build My Church".
The EO accept these passages with a similar interpretation but say Peter is
primary in honour but not
supreme or
infallible on official declarations. Catholics believe the Pope alone has the charism of infallibility meaning that when the Pope speaks
ex cathedra (in his role as sucessor of Peter and
vicarius Christi in teaching all of Christ's faithful) on matters of faith and morality, he cannot err. The EO see this as an addition to the deposit of Faith and an abuse of Petrine (of Peter) primacy.
Clear as mud? Basically the Eastern Orthodox say the successor of Peter and Patriarch of the West (i.e. the Bishop of Rome) deserves special honour but has no authoritative or adminstartive power over the other Patriarchs, nor is he the universal pastor and shepherd. He is only the first among equals with all other Patriarchs. They believe it was so since the beginning but Rome tried to exert unwarranted power over others and this was a departure from the intentions of Christ and the real meaning of Petrine primacy
Catholics say he is infallible (cannot be wrong) on matters of faith and morality when speaking in his role as Petrine successor. They belive he is the vicar (representative) of Christ, and as universal pastor and shepherd (under Christ, we're talking earthly here) has authority over all human beings on the face of the earth...no earthly power is greater than his. This includes all Patriarchs. We believe it has been so since Peter and still is so for Benedict.
Now, as to who is right...well, it's clearly the Catholics.
Not that I'm biased or anything...
Rant over. You can wake up.
Peace
Rob