Filioque is Latin for "and the Son." It is a phrase added to the official version of the Nicene Creed by the Roman Catholic Church during the period of political separation between A.D. 476, when the western Roman empire fell, and A.D. 800.
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The official version of the Creed, ratified at the council of Constantinople in A.D. 381, says that the Holy Spirit proceeds from the Father. Somewhere during that period of separation the Roman Catholics added filioque to that phrase, making it read " … proceeds from the Father and the Son."
The eastern patriarchs told the bishop of Rome that he could not do something like that on his own. He disagreed.
The discussion continued for over two centuries, until in A.D. 1054 the eastern patriarchs gave up. That division is known as the Great Schism.
Pope John Paul II publicly read the Apostles Creed, as the official version of the Nicene Creed is now known, without the filioque during his tenure as pope. This was seen as a conciliatory move by that great pope. However, nothing has come of it to this day (2009).
It was the contention of the eastern patriarchs (four of them), that all Church-wide decisions should be made by consensus of all five patriarchs. They told the bishop of Rome —the pope— that he was the leader of the patriarchs, but he could not make decisions on his own.
The popes of the period of debate, approximately A.D. 800 to 1054, disagreed. They felt that it was perfectly acceptable to add the filioque on their own authority, especially since believing that the Holy Spirit proceeds from the Father and the Son is a perfectly orthodox belief, confirmed by early fathers of the Church.
The Orthodox churches, those led by the patriarchs, make no large decisions—like changes to the creed—without a council. They believe there are seven ecumenical councils that carry authority. Because of this they could not agree with the pope.
To this day the Orthodox churches will make no important Church-wide decisions—the equivalent of the pope's ex cathedra pronouncements—because the five patriarchs are not together.
As an aside, they also believe that all western Christians should be affiliated with the pope, since he is one of the patriarchs. They see the Protestants as schismatics from the Roman Catholic Church, which they believe should be just one of the Orthodox churches.