From all eternity the Father has a perfect idea of himself, that idea is a person, it is the Son. The Son loves the Father and the Father loves the Son perfectly. That love is a person - the Holy Spirit. Three persons, one God. God is not in solitude but He is a communion of love. Each person of the Trinity empties himself out to the other in sacrifice.
Genesis points out that we are made in His Image- we see this most clearly in the family - A communion of love. The father in a family is like God the Father; the mother is like God the Son; the child is like God the Holy Spirit. The two shall become one and nine months later that one is born. They are one family yet three persons. The Family is the Smile of God!
This is why we baptize in the NAME (not names) of the Father and of the Son and of the Holy Spirit. In baptism we are given a share in the life of the Blessed Trinity - He bestows on us the gift of eternal life.