The following excerpts are taken from the Catechism of the Catholic Church regarding the Church’s teaching on the Most Holy Trinity:
The formation of the Trinitarian dogma
249: From the beginning, the revealed truth of the Holy Trinity has been at the very root of the Church’s living faith, principally by means of Baptism. It finds its expression in the rule of baptismal faith, formulated in the preaching, catechesis, and prayer of the Church. Such formulations are already found in the apostolic writings, such as this salutation taken up in the Eucharistic liturgy: “The grace of the Lord Jesus Christ and the love of God and the fellowship of the Holy Spirit be with you all.”
250: During the first centuries the Church sought to clarify its Trinitarian faith both to deepen its own understanding of the faith and to defend it against the errors that were deforming it. This clarification was the work of the early councils, aided by the theological work of the Church Fathers and sustained by the Christian people’s sense of faith.
The dogma of the Holy Trinity
253: The Trinity is One. We do not confess three Gods, but one God in three persons, the “consubstantial Trinity.” The divine persons do not share the one divinity among themselves but each of them is God whole and entire: “The Father is that which the Son is, the Son that which the Father is, the Father and the Son that which the Holy Spirit is, i.e., by nature one God.”…
254: The divine persons are really distinct from one another. “God is one but not solitary.” “Father,” “Son,” “Holy Spirit” are not simply names designating modalities of the divine being, for they are really distinct from one another…They are distinct from one another in their relations of origin: “It is the Father who generates, the Son who is begotten and the Holy Spirit who proceeds” The divine Unity is Triune.
255: The divine persons are relative to one another. Because it does not divide the divine unity, the real distinction of the persons from one another resides solely in the relationships which relate them to one another: “In the relational names of the persons the Father is related to the Son, the Son to the Father, and the Holy Spirit to both. While they are called three persons in view of their relations, we believe in one nature or substance.”…