It’s interesting, after reading the links provided by Oxy, how the Orthodox Church still dose not understand the Filique and continues to ignore the constant teaching of the Church regarding the primacy of Rome. It is very simple, in fact.
The Church teaches that the Holy Spirit proceeds from the Father through the Son, and, because He proceeds from the Father through the Son, He proceeds from the Father and the Son.
Catechism of the Catholic Church
245 The apostolic faith concerning the Spirit was confessed by the second ecumenical council at Constantinople (381): "We believe in the Holy Spirit, the Lord and giver of life, who proceeds from the Father."71 By this confession, the Church recognizes the Father as “the source and origin of the whole divinity”.72 But the eternal origin of the Spirit is not unconnected with the Son’s origin: "The Holy Spirit, the third person of the Trinity, is God, one and equal with the Father and the Son, of the same substance and also of the same nature. . . Yet he is not called the Spirit of the Father alone,. . . but the Spirit of both the Father and the Son."73 The Creed of the Church from the Council of Constantinople confesses: "With the Father and the Son, he is worshipped and glorified."74
246 The Latin tradition of the Creed confesses that the Spirit “proceeds from the Father and the Son (filioque)”. the Council of Florence in 1438 explains: "The Holy Spirit is eternally from Father and Son; He has his nature and subsistence at once (simul) from the Father and the Son. He proceeds eternally from both as from one principle and through one spiration… And, since the Father has through generation given to the only-begotten Son everything that belongs to the Father, except being Father, the Son has also eternally from the Father, from whom he is eternally born, that the Holy Spirit proceeds from the Son."75
247 The affirmation of the filioque does not appear in the Creed confessed in 381 at Constantinople. But Pope St. Leo I, following an ancient Latin and Alexandrian tradition, had already confessed it dogmatically in 447,76 even before Rome, in 451 at the Council of Chalcedon, came to recognize and receive the Symbol of 381. the use of this formula in the Creed was gradually admitted into the Latin liturgy (between the eighth and eleventh centuries). the introduction of the filioque into the Niceno-Constantinopolitan Creed by the Latin liturgy constitutes moreover, even today, a point of disagreement with the Orthodox Churches.
248 At the outset the Eastern tradition expresses the Father’s character as first origin of the Spirit. By confessing the Spirit as he “who proceeds from the Father”, it affirms that he comes from the Father through the Son.77 The Western tradition expresses first the consubstantial communion between Father and Son, by saying that the Spirit proceeds from the Father and the Son (filioque). It says this, “legitimately and with good reason”,78 for the eternal order of the divine persons in their consubstantial communion implies that the Father, as “the principle without principle”,79 is the first origin of the Spirit, but also that as Father of the only Son, he is, with the Son, the single principle from which the Holy Spirit proceeds.80 This legitimate complementarity, provided it does not become rigid, does not affect the identity of faith in the reality of the same mystery confessed.
The Church teaches that Peter was given primacy, and, as such, the Roman church, the Church in Rome, the Bishop of Rome, who is the Pope, is head of the whole Church.
See that ye all follow the bishop, even as Jesus Christ dose the Father. He who honors the bishop has been honored by God; he who dose anything without the knowledge of the bishop, dose serve the devil. - St. Ignatius, bishop of Antioch 107 AD
When we refer them to that Tradition which originates from the Apostles, which is preserved by means of the succession of presbyters [bishops] in the churches, they object to Tradition, saying that they themselves are wiser, not merely than the presbyters, but than even the Apostles. - St. Irenaeus, bishop of Lychos 180 AD
It is necessary to obey the presbyters who are in the Church, those who, as I have shown, possess the succession from the Apostles. - St. Irenaeus
For it is with this Roman church, by reason of its more powerful pre-eminence, that every church, that is to say all the faithful everywhere, ought to agree. - St. Irenaeus
Indeed, the others were that also which Peter was; but a primacy is given to Peter, whereby it is made clear that there is but one Church and one chair. - St. Cyprian, bishop of Carthage 251 AD
There are many other things which most properly can keep me in [the Catholic Church’s] bosom. The unanimity of peoples and nations keeps me here. Her authority, inaugurated in miracles, nourished by hope, augmented by love, and confirmed by her age, keeps me here. The succession of priests, from the very see of the apostle Peter, to whom the Lord, after his resurrection, gave the charge of feeding his sheep, up to the present episcopate, keeps me here. And last, the very name Catholic, which, not without reason, belongs to this Church alone, in the face of so many heretics, so much so that, although all heretics want to be called ‘Catholic,’ when a stranger inquires where the Catholic Church meets, none of the heretics would dare to point out his own basilica or house. - St. Augustine, bishop of Hippo 397 AD
If the very order of episcopal succession is to be considered, how much more surely, truly, and safely do we number them from Peter himself, to whom, as to one representing the whole Church, the Lord said, ‘Upon this rock I will build my Church, and the gates of hell shall not conquer it.’ Peter was succeeded by Linus, Linus by Clement. - St. Augustine, 412 AD