The Western church says the Holy Spirit "proceeds from the Father and the Son". The Eastern churches say "from the Father". In Latin the difference is one word, filioque, "and the Son". It has divided Christians for over a thousand years, and it was never only about the word.
That the great split of 1054 between Rome and Constantinople was caused by this one word. It was one grievance among several, alongside the authority of the pope, the kind of bread used at communion, and centuries of growing apart. In fact the papal legate's bull of excommunication in 1054 accused the Greeks of having removed the word, which they never had.
That it is an argument about nothing. Even people who think it should never have divided the church agree that it raises two real questions.
The theology. The Creed agreed at Constantinople in 381 says the Spirit "proceeds from the Father", echoing John 15:26. The Latin tradition, following Augustine, added that the Spirit also proceeds from the Son. It did so to show that the Son is fully equal to the Father, and to explain how the Son and the Spirit are distinct. The Eastern tradition held that the Father alone is the source within the Trinity, and that the Latin formula blurs the Father's role and makes the Spirit seem secondary. Both sides agree that the Son sends the Spirit into the world (John 15:26; Galatians 4:6). The dispute is about the eternal life of God.
The authority. The Creed belonged to the whole church, agreed by an ecumenical council, and the Council of Ephesus (431) forbade composing a different creed. When the West added a word without the East, the East asked a question that went beyond doctrine: can one part of the church change the common creed on its own?
Each stated as its holders would have stated it.
The Latin West, following Augustine's On the Trinity
The Spirit proceeds from the Father and from the Son, though principally from the Father, who gives the Son everything he has, including this.
Why they held it Because whatever the Father has, he gives to the Son (John 16:15), and because Arian groups in Spain were still denying that the Son was equal to the Father.
Photius of Constantinople and the Eastern Orthodox tradition
The Father is the one source within God. The Son is begotten of the Father, and the Spirit proceeds from the Father. The Son's sending of the Spirit belongs to his work in the world.
Why they held it Because two sources within God would undermine his unity, because John 15:26 says "from the Father", and because the Creed was not the West's to change.
Maximus the Confessor; the Council of Florence (1439); many modern theologians on both sides
The two traditions can be reconciled: the Spirit comes from the Father as the only source, through the Son.
Why they held it Because some Greek fathers already used "through the Son", and because the two sides may mean more compatible things than their formulas suggest.
The word arrived in the Creed slowly, region by region, and Rome was among the last to say it.
Because it shows that how a church decides can matter as much as what it decides. Many Christians today can say both formulas in good conscience. Far fewer can explain why a creed agreed by everyone should not be changed by one side, however right that side thinks it is.