What is the Trinity, and how can God be three and one?

The teaching. "The Trinity is One. We do not confess three Gods, but one God in three persons" (CCC 253). Father, Son, and Holy Spirit are one in substance (what God is) and three in Person (who God is). Each Person is wholly God; they are distinct in their relations to one another; they are never separated (CCC 253-255). This is "the central mystery of Christian faith and life" (CCC 234).

Not a contradiction. God is not three and one in the same respect. He is one nature and three Persons. A contradiction would be "three Gods and one God." The Church is careful to say neither that (tritheism) nor that the three are just masks of one Person (modalism). We are not saying 1 + 1 + 1 = 1. We are saying that in God, being and relationship are one reality in a way our minds cannot fully picture but can see is not absurd.

Where it comes from. Jesus commanded Baptism "in the name of the Father, and of the Son, and of the Holy Spirit" (Mt 28:19), one name, three Persons. At His Baptism the Father speaks, the Son stands in the water, the Spirit descends (Mt 3:16-17). Jesus says, "The Father and I are one" (Jn 10:30) and "Before Abraham came to be, I AM" (Jn 8:58). St. Paul blesses in the name of all three (2 Cor 13:13). The Councils of Nicaea (325) and Constantinople (381) defended it against Arius, giving us the Creed.

Why it matters. Because God is a communion of Persons, "God is love" (1 Jn 4:8) is literally true from all eternity, not just since creation. Love is what God is. And we are invited into that love: the Son became man so that in Him we might become sons and daughters of the Father, filled with the Spirit (CCC 260). Every Sign of the Cross is a profession of the Trinity.

Analogies (all imperfect): St. Patrick's shamrock, water as ice, liquid, and steam (careful: that leans modalist), St. Augustine's memory, understanding, and will in one mind. Use them as ladders, then leave them behind.