This is called the doctrine of the Trinity. It teaches that the one God acts as three persons, in three kinds of ways, but always as one God. 

But the belief in God as Trinity does not mean the persons divide the work between them: no one person of the Trinity ever acts alone. When one acts, all act together, because God is one. 

Whenever we read of the action of God, it is always the three persons acting together. There are very many examples of this. Here are just ten: (1) At the creation, the Father spoke his Divine Word and created through the power of the Spirit. (2) The Father breathed his Spirit into Man to make him in his own image. (3) The Spirit spoke God’s Word through the prophets. (4) The Angel visited Mary with the message of the Father so that she conceived the Word by the power of the Spirit. (5) St John tells us ‘the Word (with God from the beginning) became flesh and dwelt amongst us’. (6) At his Baptism, the Spirit descends on Jesus like a dove as the Father speaks from heaven. (7) At his Crucifixion, the Son says ‘Father, into your hands I commend my Spirit’. (8) And when the Son ascends to the Father he sends the Spirit, the Advocate, the Paraclete, to lead us into all truth. (9) We pray to the Father, through the Son, in the Spirit. (10) At Mass, we pray to the Father that he send the Spirit on the bread and wine that they become the body and blood of his Son. 

Again and again and again it is Father, Son and Spirit who move together, who operate together. 

And notice that in all these examples I give, and in many others, the movement of the Trinity is about entering into the created world, about reaching to humanity. It is about involvement, action. The Trinity is about drawing human beings into the life of God, in creation, in redemption, in prayer, in sacraments. The best description of all is of course love - because love always involves another, and God is love because he lives love in himself, and extends his loving hand, his loving word, his loving Spirit, to embrace all humanity. 

Each person of the Trinity, Father, Son and Holy Spirit, is fully God, so prayer can be addressed to any one of the persons and to all three together. 

However, some prayers might be more appropriate to be made to a particular person. 

For example, in the liturgy we take part in the action of Christ as high priest, so our prayers are usually to the Father, through the Son and in the unity of the Holy Spirit. Notice that in the Mass prayers nearly always are phrased in this way. It is quite unusual to address a prayer in the liturgy directly to Jesus or to the Holy Spirit. 

However, private prayers or other acts of devotion may be different. 

For example, if we are praying for God’s mercy, we may often speak directly to the Son, especially to the Sacred Heart, which is the expression and example of the love and mercy of God. 

Similarly, we would naturally pray to the Holy Spirit, if we are praying for grace, for the power of Divine Love, or for the many gifts or fruits which come to us from the work of the Spirit in our lives.