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Augustine divides his discussion into fifteen books, which can be roughly summarized in the following ways, if we see it as a route from Faith to (hope of) Understanding. As he repeatedly says, quoting a translation of Isaiah, "Unless you believe you will not understand"; and he proposes a rule for inquiry in these matters in Book VIII: to preserve by firmness of faith what has not yet become clear to understanding.
Books I-VIII: The Doctrine of the Trinity as Received by Faith
Book I: The unity and equality of the Three Persons is shown from Scripture.
Books II-IV: The unity and equality of the Three persons continued; in particular, the missions of the Son and the Spirit do not indicate any inferiority.
Book V: Why the Father's being unbegotten and the Son's being begotten does not indicate a difference of nature: not everything predicated of God is predicated according to substance, since some things are predicated relatively, either with mutual reference (as Father and Son) or by relation to creature (as Creator or Lord).
Book VI: What is meant by saying that Christ is the wisdom and power of God
Book VII: The wisdom and power of God, continued; comparison of Latin and Greek modes of expressing the unity and distinction of the Trinity
Book VIII: A problem: we cannot know God without loving Him, but what likeness can we find to help us believe Him that we may love Him enough to come to know Him? Love itself has a trinitarian character: the lover, the beloved, and the love that unites them. And we are told by Scripture that God is love. Thus we must love others, and in our love we can see and love love itself and by reflection in it see the Trinity.
Books IX-XV: The Doctrine of the Trinity as Reflected in the Image of God
(Augustine trains the reader in things that are made in order that they may know the one by whom they were made.)
Book IX: A trinity in the created mind: mind, self-knowledge, self-love
Book X: Another, more manifest, trinity in the created mind: memory, understanding, love. There are a number of complications with this trinity, however, not least that the mind can be said to remember, understand, and love itself even when not thinking itself, and that the mind when it thinks itself does not always clearly distinguish it from the body. The trinity will be set aside for a moment in order to clarify certain aspects of human nature.
Book XI: Imperfect trinities in the 'outer man': the object seen, exterior vision, the purpose of will (intentio) that combines the two; also sense-memory, internal vision, and will. (This book often strikes people as an odd digression. But Augustine is preparing for his discussion of the trinities of the inner man, and he will explicitly use the discussion of this book in this way in Book XIV.)
Book XII: Returning to the 'inner man', since we look not merely for a trinity but for an image of the Holy Trinity, we must determine what is meant by 'image of God': the true image of the Trinity can only be found in that part of the human mind that contemplates eternal things. Properly, this is the superior reason, to which wisdom (contemplative aspect of human life) pertains; but the inferior reason, to which knowledge (active aspect of human life) pertains, has some relation to it. The inferior reason, however, considers temporal things.
Book XIII: The trinity of the inner man according to the inferior reason, which is a trinity of faith; we need faith, both a temporal faith in eternal things and a temporal faith in temporal things like the life of our Lord, to reach the beatitude we all will to have.
Book XIV: The trinity of the inner man according to the superior reason. The trinity of faith, which will pass away, cannot be the image of God; nor can the trinity that will replace it when it does. Rather, the image of God must lie in that aspect of us which may partake of God Himself. In the mind remembering itself, knowing itself, and loving itself, we find a trinity that is the image of God (albeit one that is impaired and disfigured by sin) insofar as such a mind is capable of remembering, knowing, and loving God. This trinity is renewed by grace; thus faith, by which we receive grace, remains important to it.
Book XV: The trinity of the inner man according to the superior reason, continued. From the image of the Holy Trinity we wish to arise to the Holy Trinity itself: this leads us immediately to the inadequacy of any created trinity for understanding the Uncreated Trinity. We see by way of a mirror, in an enigma. He discusses the image of God found in this state of enigma in greater detail, showing its likeness and unlikeness to the Holy Trinity. The image of God, however, will be renewed; and then in that beatitude we will be like God, seeing Him not in a mirror but face to face, as He is.
Some important points that are usually forgotten or ignored:
1. It is very important to grasp that Augustine is not engaging in random speculation or just discussing the Trinity for the sake of discussing the Trinity. He has opponents in view. De Trinitate is an anti-Arian work, broadly speaking. He first lays out what the doctrine of the Trinity is, and how it is arrived at. And then he considers the different trinities in the last stretch of the book specifically to address to the question of how we can understand the doctrine of the Trinity well enough at least to believe it, and not just be repeating the words. And that ties to the second point, because his answer is that by charitable love we live the reflection of the Holy Trinity.
2. One remarkable feature of Augustine's discussion that is often overlooked is that the whole point of the second half of the work is to lay out how we can "live the trinity of the inner man" as expressed in wisdom. The whole discussion is geared to clarifying what it means to live a life in light of the Holy Trinity. It is not an abstract discussion about an abstract doctrine, but an inquiry into the Christian mode of life.
3. Strictly speaking, Augustine does not think the image of God in man is the mind remembering, knowing, and loving itself. In fact, he explicitly denies this in the ordinary sense. The image of God in man is the mind insofar as it is capable of remembering, knowing, and loving God. That is, the basic image of God in us is our capability for worshipping God, which begins to make us wise; and it is in wisdom that we find the trinity that can properly be called an image of the Holy Trinity. The connection between the two, of course, is that on the Christian view genuine love of self and love of God go hand in hand; we can only love ourselves (and thus remember and know ourselves) rightly if we love (and thus remember and know) God, and love (and remember and know) ourselves in light of Him. Life in the image of God is fundamentally a life of loving God, and, in loving God, loving our neighbor as we love ourselves in God.
4. The distinction between inferior reason and superior reason, while extremely important, complicates the discussion considerably, more than is generally recognized. The inferior reason by itself cannot have an image of God, because it is concerned only with temporal things. But the inferior reason and the superior reason together are the one human mind and the one image of God.
5. Throughout the discussion of the various created trinities, Augustine is not looking merely for triads, but for triads that exhibit the following characteristic, at least in some sense: each is in each, each in all, all in each, all in all, and all are one.
6. De Trinitate ends with a prayer essential to understanding the argument of the work; the last part of which is this:
O Lord, the One God, God the Trinity, whatever I have said in these books as coming from You, may they acknowledge who are Yours; but if anything as coming from myself, may You and they who are Yours forgive me. Amen.
