We'll start with water, something everyone has experienced. As we all know, water can be found in one of three states: gas, liquid, or solid. Although each of these three states contains exactly the same molecules at exactly the same ratios, the behavior and characteristics of each is completely different. In fact, one would almost deny that they were the same substance at all if the evidence didn't prove otherwise!
So it is with the Trinitarian God. It is one God, with three "states" or "persons": the Father, the Son, and the Holy Spirit. These persons each have unique, definable characteristics that make them distinct enough from one another that there are many who have been confused into believing they are separate deities. The primary Characteristic of God the Father is justice, the primary characteristic of God the Son is mercy, and the primary characteristic of God the Holy Spirit is hope. All three of these are aspects of God who is Love, but these aspects are different enough to make them appear separate.
Here is the Catechism of the Catholic Church
261 The mystery of the Most Holy Trinity is the central mystery of the Christian faith and of Christian life. God alone can make it known to us by revealing himself as Father, Son and Holy Spirit.
262 The Incarnation of God's Son reveals that God is the eternal Father and that the Son is consubstantial with the Father, which means that, in the Father and with the Father the Son is one and the same God.
263 The mission of the Holy Spirit, sent by the Father in the name of the Son (Jn 14:26) and by the Son "from the Father" (Jn 15:26), reveals that, with them, the Spirit is one and the same God. "With the Father and the Son he is worshipped and glorified" (Nicene Creed).
264 "The Holy Spirit proceeds from the Father as the first principle and, by the eternal gift of this to the Son, from the communion of both the Father and the Son" (St. Augustine, De Trin. 15, 26, 47: PL 42, 1095).
265 By the grace of Baptism "in the name of the Father and of the Son and of the Holy Spirit", we are called to share in the life of the Blessed Trinity, here on earth in the obscurity of faith, and after death in eternal light (cf. Paul VI,CPG § 9).
266 "Now this is the Catholic faith: We worship one God in the Trinity and the Trinity in unity, without either confusing the persons or dividing the substance; for the person of the Father is one, the Son's is another, the Holy Spirit's another; but the Godhead of the Father, Son and Holy Spirit is one, their glory equal, their majesty coeternal" (Athanasian Creed: DS 75; ND 16).
267 Inseparable in what they are, the divine persons are also inseparable in what they do. But within the single divine operation each shows forth what is proper to him in the Trinity, especially in the divine missions of the Son's Incarnation and the gift of the Holy Spirit.