The Catholic and Orthodox Churches, as well as most Protestant communities, have taught for generations that God is  “incomprehensible and ineffable.” This means that God is so wholly other – so unlike anything we have experienced in the material world – that we are incapable of fully or perfectly expressing what he is like in his essence or nature.
For this reason, when we attribute to God certain qualities – wisdom, omniscience, omnipotence, etc. – we are not saying that he has them in addition to his being, as we would when we attribute wisdom to Mr. Jones (who would still remain Mr. Jones even if he lost wisdom). We mean that if God were a composite being like Mr. Jones – having accidents distinct from his substance – then God could not be the First Cause, since anything that is composed requires a cause to do the composing.
As St. Thomas notes , “[E]very composite has a cause, for things in themselves cannot unite unless something causes them to unite.” So God must be “simple.” Any attributes we ascribe to God (wisdom, omniscience, omnipotence, and so forth) cannot literally be “parts” or “accidents” of God. This is why we say that God is identical to his attributes and his attributes identical to each other (a hard thing to picture, but true).
Unlike God’s simplicity, which can be known by philosophical reflection, the doctrine of the Trinity is known by revelation. Although Scripture teaches that there is only one God by nature – the self-existent source of all contingent beings (Acts 17:24-28) – we learn that he is Father, Son, and Holy Spirit as well. So within the simplicity and unity of God, there are three divine persons, distinct from each other, though each is fully God.
The Church’s task cannot be to explain the Triune God, for it cannot be done. Rather, the best we can hope for is to show the doctrine to be “thinkable without contradiction,” as Brian Davies, O.P.  puts it. Given the ontological gulf between Creator and creation, we are confined to using analogies from the created order.
Scripture teaches that the Son is the Word generated (or begotten) from God (Jn 3:16). And yet the Word is God (Jn 1:1-2) and there is only one God by nature. How can that be? Consider an analogy. A word or thought is generated by the mind. So, in a sense you are paternally related to your thought insofar as you generate it. And your thought is filially related to you insofar as it is generated by you.
But a thought is not something that is external to you, as it would be in the case of someone physically generating a child or writing a sentence on a piece of paper. The relation between me and this thought is a relation internal to my being. Nevertheless, there is a real relation, a distinction, between me and the concept of me. Suppose, for example, the thought I generate is a concept of myself. Although this thought is an “accident” that inheres in me and does not share my nature, it is similar to me because it is a thought about me.
Now let’s apply this reasoning to God’s internal life. God has a concept of Himself. Because He is eternal, simple, and perfect, this thought of Himself must be eternally generated, perfect, and not an accident. In contrast, my thought of myself, though inhering in my mind, is generated in time, imperfect, and an accident.
Thus, God’s thought (or Word) of himself must express the fullness of the divine nature: eternality, simplicity, perfection, etc. But this means that the Word, eternally begotten, is not only internal to God’s nature, he is God, just as the Father, the eternal begetter, is God.
As the great Frank Sheed once put it : “[T]here is a huge difference between God’s Idea and any idea we may form. His is Someone, ours is only something.” Because God is eternal and simple, there is only one Word (Jesus Christ), one act of intellect, from which creation derives its order and being. (Cf. Col 1:15-17)
Scripture teaches us that the Holy Spirit proceeds from both the Father and the Son (Jn 15:26), and the Holy Spirit is in fact God. (Acts 5:1-6) Assuming that God’s external relations to the created order – as noted in these passages – model his immutable internal relations, the Holy Spirit eternally proceeds from both the Father and the Son. The Son, as we have seen, is the fullest expression of the divine intellect. So the Spirit must be the fullest expression of the divine will.
The Spirit, as the Church teaches, is the eternal and unchanging love that the Father and Son have for each other. But just as God’s perfect exercise of intellect, the thought of himself, must be eternally generated, not an accident, and perfect, God’s perfect exercise of will, the love that the Father and the Son have for each other, must be eternally spirated, not an accident, and perfect as well.
But this means that the love eternally proceeding from the Father and the Son is not only internal to God’s nature, he is, like the Word, a Divine Someone, just as the Father is a Divine Someone.
As hard as these notions may be to grasp, the distinctions between the persons of the Trinity are grounded in real relations internal to God, which are eternal and unchanging: the Father begets the Son, the Son is begotten, and the Holy Spirit proceeds from the Father and the Son.
Moreover, since there is only one divine nature, and that nature is simple, each person of the Trinity must fully share in the divine nature. There is a mutual indwelling of the divine persons that follows from divine simplicity. It is, to be sure, a mystery, one that goes beyond the ability of the human mind to fully comprehend. But it is not wholly unthinkable.