1. We begin with God. “God" is the name we give to that being which is the creator of everything else that exists. By nature we can know that there is such a being, but we cannot know its characteristics except indirectly by reasoning from what is seen in its effects (this is called the "way of remotion"). Mostly this involves denying that God could have certain characteristics: e.g. mutability, finitude, imperfection, etc.
2. As Catholics, we believe that God created rational animals (humans) for the purpose of inviting them to participate in his supreme perfection: to know as he knows, to love as he loves, to delight as he delights.
3. All humans by nature are capable of knowledge, but only to the extent that our senses allow it. Our minds are limited by what they can receive from the world, and since we receive information through our senses, what we know by nature is necessarily limited to the material, sensible world around us
4. God however is not a material, sensible being, in his essence, and therefore to know the essence of God is impossible for humans by nature, because we cannot see or perceive God.
5. God, then, desiring the supreme perfection of our intellectual capacities, the natural object of which is the true and the good, stretches us beyond what we could naturally learn to something that exceeds material reality... i.e., to spiritual realities.
6. And he does this so that we can come to know Him, and, knowing him, become friends with Him, and by sharing in the understanding and love which he has of himself and of all things, to participate in the supreme perfection which is the Divine Life.
7. The primary way in which God has made himself known to humanity is Jesus Christ, in whom God, the eternal one, joined himself to human nature, and, as St. John says, dwelt among us.
8. But Christ came not just to express the love of God, but also to make God known to us, and one of the things Christ reveals about the Godhead is its essential relationality, i.e. that God subsists as an eternal relation among three persons: one innascible, one begotten, one proceeding.
9. We can explain what these relations mean by an analogy to the human acts of knowing and loving. This is a faint and imperfect analogy, but it helps somewhat.
10. In the human act of knowing, the essence of what is known, i.e. what the thing is that we are trying to understand, comes to be abstractly in the mind of the one knowing. I look at a flower, I study the flower, and what it is to be the flower, the flowers characteristics and essence, are situated in my mind, so that I possess them mentally independent of the flower.
11. The human act of knowing is limited, however, because (1) we only ever understand things finitely and imperfectly, and (2) the objects of our natural knowledge are material, and a material thing cannot really come to exist in the mind, since the idea held by the mind is immaterial.
12. Now, consider God's act of knowing Himself. In God there is no finitude of understanding, and since the object of that understanding is not material, the idea God has of himself, the inner word by which he refers to himself, is supremely perfect. So perfect, that it has the fullness of the essence and reality of the Divine Nature. In other words, within God as knower, the idea of God is fully Divine.
13. The act of understanding terminates in a being, an idea, which is God in essence and existence, but in one who is, while sharing in the very same act of existence and fundamentally inseparable, distinct by the relation of knower to known.
14. This is what theologians refer to as "the procession of the Word”. (The Word being Christ, the second person of the Trinity.)
So much, then for the analogy to knowing. Now, the analogy to loving.
15. In the act of knowing something, we judge it to be good or deficient, and insofar as we judge something to be good, our will is naturally moved toward the object under consideration, which we desire or love. And this impulse of the will proceeds from the subject as a whole through the idea of what is known, as an act of love which participates in the idea of what is known.
16. In God, again, love being infinite and understanding being utterly perfect, that impulse toward the goodness God finds in his own supreme being, is so full of reality and so unlimited that it again attains to the fullness of what is loved, which is God, and subsists as what is God, and thus is a person, inseparable from the ones from which it proceeds, but nonetheless really distinct by its relation to them, which we call the Holy Spirit. The Spirit is the impulse of love which proceeds from the Father and the Son through the eternal self-knowledge of God.
17. Now, one more thing before we finish. In Catholicism, we talk about the persons of the Trinity, i.e. these three subsistent persons within God, in two ways.
18. First, we talk about them as they exist eternally as the three distinct faces of the Godhead. Second, we talk about them as associated with certain acts of God with respect to creatures.
For example, we normally refer the act of creation principally to the Father, since the Father is the innascible person in the Trinity, the "source" so to speak.
19. We refer the act of redemption to the Son, because the Son united himself to human nature and offered himself as a sacrifice in atonement for our sins.
20. We refer the act of sanctification to the Spirit, since the Spirit is the fruit of the love within the Godhead, and sanctification is marked by an increase in the theological virtue of charity, and is a direct expression of God’s own charity toward us.
21. On these topics there's a lot to say. But some sense of that you can find by reading your Catechism.