I shall state my thesis up front - The Trinity is a short-hand for the Christian understanding of how the God we encounter has chosen to reveal who this God is.
It's really that simple.
Now comes the hard part.
Part of the problem with any attempt to unpack trinitarian thought is the western Christian attempt to make of it something rational. It is not. There is nothing rational about the claim that God is three persons in one, one substance revealed in three always-present distinct persons, wholly separate yet never separated. Unraveling the knot of Trinitarian thought usually leaves only a few alternatives, most of them unattractive to a contemporary Christian. Falling back on the Nicaean-Constantinopolitan formula, usually referred to with reference to the Greek affirmation of Son and Father being "of one substance" (homoousios), with the procession of the Spirit added as an afterthought (at least in the west, where the procession of the Spirit became a bone of contention culminating in an official split with the Eastern churches in the eleventh century).
In defense of the original language of Nicaea, those present at that first, Imperially-decreed Church Council, had little else to use but the tools of neo-Platonism with which to work. In an attempt to clarify their understanding of the newly enthroned Imperial God's interaction with the world and people who were now officially Christian, they had no recourse but to use the technical vocabulary of substance and accident to express their insistence that, in the life and ministry, death and resurrection of Jesus of Nazareth, humanity encountered not just one among many possible Holy Men, a wonder-worker of both wisdom and sagacity, but something more, something, indeed, final and complete. In this executed Galilean miracle worker, once a rebel against an Empire who now demanded all citizens become believers in his message, the bishops gathered at Constantine's insistence made as clear as they could that Jesus was the Christ because, in him, humanity came to see and hear God, completely, unblemished, without any question.
All "orthodox" theology, at least as that term is generally understood, is nothing more than an attempt to distinguish itself from those theologies that make claims that do not seem to embrace fully and completely the reality of the Divine-Human encounter in Jesus of Nazareth. As early as St. Paul, we see the beginnings of the attempt to claim that in Jesus, we Christians have encountered the fully-realized exposure of who God is. Who God is can best be seen by understanding what this God does in Jesus Christ.
By the time of Nicaea, over three centuries of controversy, argument, morphing of belief, and exposure to a vast array of settings and contexts, had rendered discussions of Jesus and God, of who Jesus was and who his "Father" is incomprehensible to the original bearers of the message known originally only as "The Way". The big controversy between those who declared (according to detractors; the writings and teachings of Arius are lost to us, known only through the sarcastic recollections of his detractors) "there was a time when He was not" and those who insisted that the Divine-Human encounter made such a statement not only erroneous, but dangerously so, led the young Emperor to decide that a clearing of the muddle was necessary. The irony, at least from an historic perspective, is that the Arians, with their high vision of Divinity which rendered any talk of a fully-divine Jesus of Nazareth intellectually incomprehensible, were by far the large majority of Churchmen. Constantine's Christian mother was an Arian, as were many of his advisers. Yet, once the council was called, Arius' rivals managed to out-maneuver him; with the blessings of Imperial power, the claim that Jesus was somehow not-quite-God was not just declared wrong, but dangerous.
In the 1700 years since that time, the formula adopted at Nicaea, and affirmed over a series of councils in the century following that culminated at Constantinople, became less a touchstone upon which the Church could build a dialogue about who this God is we claim to encounter in Jesus of Nazareth, and more a relic of a dead age. Ensuing centuries saw the attempt to make sense of this declaration of God's threeness and oneness, and for the most part they lose themselves in far too much technical discussion, losing sight that, at its heart, the declaration that, as St. Paul wrote in 2 Corinthians, God was in Christ reconciling the world to Himself, lies at the very heart of Trinitarian thought. When we affirm the reality of the Trinity - Father, Son, and Holy Spirit - we are affirming that the God we Christians declare has claimed us in a unique way through Jesus Christ has chosen to reveal not something of the "whatness of Divinity" so much as something of the Divine mind and plan.
Protestant thought originally followed the scholastics in getting lost in all sorts of technical discussions concerning substance and accident, procession and interpenetration. By the beginning of the 19th century, however, there was a general recognition, at least by some, that far too much of this talk had become mere noise, a banging gone and clanging symbol. Schleiermacher's Glaubenslehre disposed of the Trinity easily enough, adding it as an appendix, a mere afterthought. As late as the mid-20th century, another Reformed theologian, Emil Brunner, could declare in his own dogmatic musings, that the Trinity is not only not Biblical, but the affirmation of the Trinity was just not all that important. It is certainly not essential to salvation (which is really beside the point).
Karl Barth, on the other hand, returned Trinitarian thought to pride of place, putting it at the very beginning of his Church Dogmatics. Yet, over a century of neglect and ridicule had rendered even Barth's heroic attempt to revive the centrality of trinitarian language and thought somewhat off-balance. He tried to speak of the "persons" as "modes of Divine revelation", which could lead the casual reader (if he had any) to think that Barth was falling back on a time-worn error.
This is nit-picking. Because of his prominence, Barth's reversal of Schleiermacher's sidelining of the Trinity opened the door to a revival of sorts of Trinitarian thought in western Protestantism*. While they have varied in success and clarity, the return of the claim that we have to do with something unique in the God Jesus called "Father", and whom we declare has claimed us through the power of the Spirit at the very least offers the possibility of returning talk about God, about revelation, even about Jesus and the life in and of the Holy Spirit a little more coherent, if not always acceptable to a world where such talk no longer has much traction.
Which returns us full circle, I believe, to my thesis. The Trinity has always been nothing more (and nothing less) than a short-hand way of describing the Divine-Human encounter we Christian claim as our unique inheritance. From it flow all the discussion we will subsequently have - on who God is, on who Jesus is, on what it means to be human in Christian terms, on the Divine plan for humanity. No real grasp of my own approach to Christian confession comes without understanding this: I am a Trinitarian by choice, because in the declaration of God as Father, Son, and Holy Spirit, we declare the full story of God and creation in the simplest possible terms. Just as working through Scripture is always the starting point for real Christian thought, so, too, is the Trinity.
*I qualify this statement with "west" because the eastern churches have never lost their Trinitarian focus. Indeed, they are so relentlessly Trinitarian, they very often verge on a kind of tri-theism which is far preferable to the watered-down, muddled unitarianism and even Divine utilitarianism of the western churches.
