A Brief Explanation of the Trinity - If there are three persons, why are there not three Gods?
The co-equal and distinct persons of the Father, the Son, and the Holy Spirit are not three Gods because they share the same essence in complete unity. This is difficult to grasp because we have no comparison that we can experience empirically or with our five senses.
For example, the patriarchs Abraham, Isaac, and Jacob are three distinct persons and beings. They do not share the same essence or being or divine attributes. However, the doctrine of the Trinity says that the Father, the Son, and the Holy Spirit share the same being or essence, and this is where the human example of the three patriarchs and the divine nature must part company. So the human analogy fails to describe the Trinity and would in fact confuse this doctrine if we took the example seriously.
Augustine comes up with an analogy that is deep: the lover, the beloved, and love (The Trinity, Books VIII.14; IX.2, and XV.10). As we saw at the baptism of Jesus, the Father says that he loves the Son, and at that very moment, the Spirit descends and rests on Jesus (see no. 1, above, and Matt. 3:16-17). Augustine seems to say that the Father loves the Son, who receives his love and returns it, for example, in his willingness to obey the Father and die on the cross, while the Spirit communicates the love between them. This image of a triad of love expresses how Christians believe the Father, the Son, and the Holy Spirit existed before the creation of time and the universe, and how the three persons will forever exist. It is into this love that the Trinity welcomes all believers and all who have received the love of God, through Christ and by the power of the Holy Spirit.
Augustine’s analogy is profound, because we have all experienced love from a lover, and we have returned the same love. But it still does not adequately explain how the one God shares this divine attribute of love and all the other divine attributes in three persons. Our human love is but a poor reflection of divine love. We cannot observe this with our eyes, so all analogies from nature and humanity ultimately fail in illustrating the Trinity. Analogies from nature cannot reach up to heaven so that we can connect them to the Trinity. Our finite and limited mind cannot grasp this mystery, even after our minds have studied the doctrine for years.
The strict unity of God does not make sense in terms of the love of God because divine love amounts to self-love in a strict oneness doctrine of God.
To clarify further why there are three persons, but not three Gods, God’s essence is not divided equally into three parts; it is not one-third, plus one-third, plus one-third. Also, the three persons are not added on to God’s essence or being, as if they are tacked on to the outside of God. Rather, all three persons share the same essence, fully God in one being, in total and perfect unity.
Thus, basic Christian doctrine teaches that one God exists in three persons: the Father, the Son, and the Holy Spirit, not in three Gods, which is called tritheism.
Besides giving a list of divine attributes, this website uses some properties of creation and this series presents a mathematical model to illustrate the Trinity, though, as noted, all analogies are ultimately flawed.