Yesterday’s Solemnity of the Most Holy Trinity is the celebration of a mystery: that three persons are somehow in one God. It took time for Church to fully understand the triune nature of God, drawing out the trinitarian references from scripture and trying to construct a deeper philosophical and theological context for understanding them.
[Since this is a long theological essay, I've placed the rest after the jump.]
In John 8:42, Jesus tells us something new: something that would challenge the Church Fathers to find a new understanding of the substance and person of God. “If God were your Father,” he says, “you would love me, for I proceeded and came forth from God; I came not of my own accord, but he sent me.” It is one of the most difficult truths in all Christianity: how three can be one, and one three. This profound mystery is so deeply imbedded in both scripture and tradition that it remains one of the defining elements in Christian theology.
Jesus came to reveal the true nature of God, and it became abundantly clear, from the evidence of scripture and the earliest beliefs of the Church, that this nature is triune. When the apostles are left with their commission at the end of the Gospel of Matthew, Jesus tells them to “Go therefore and make disciples of all nations, baptizing them in the name of the Father and of the Son and of the Holy Spirit.” (Matt 28:19) Even at this early date, the Trinitarian formula is clear.
The question of how these three persons relate one to another, and how they form a unity, was shaped in the earliest centuries of the Church. Contact with obvious heresies—such as Arianism and Modalism—sharpened the understanding and arguments of the Fathers, as they were forced to develop a clear doctrine explaining how a single divine essence could somehow contain three diverse persons. It was by understanding and explaining the idea of processions that these ideas and relationships came into sharper focus.
Although both Eastern and Western Fathers wrestled with the precise nature and terminology of the processions, it was St. Augustine who first codified it for the Church, and St. Thomas Aquinas who added a more precise terminology.
St. Augustine made clear that the substance of God was prior to the persons, although the idea of “prior” suggests a temporal element that does not apply to God in the same way it applies to humans. (Augustine acknowledges the limitations of temporal language when he observes that “it is easier for speech which has to proceed in a time dimension to explain something which is comprehended in the time dimension.”) Later, St. Thomas would write, “All that exists in God, is God.” 
The Father is the principium, the first among equals who initiates the processions. A “Father” is someone who “communicates life and existence to another.” There are two kinds of “creation” in God. (We shall explore the limitations of the word “creation” in the following section.) The first is an external process that results in something outside of God. The second is an internal process of knowing and willing by the substance of God.
One is “changeable nature,” while the other is “unchangeable truth.” The creation of man and the material universe is an external process of creation, while the procession of the Son and the Holy Spirit are an internal process by which the single substance of God communicates the divine essence to the other two persons of the Trinity. To better understand this, we must separate the processions into two terms: the “generation” of the Son and the “spiration” of the Holy Spirit.
The relationship of the Father to the Son is self-evident in their titles: a son comes from a father. A mortal father creates his progeny through an act of love with the mother. The Son of God, however, is not “created” in the same way. He is eternally begotten by the Father alone, “God from God.” Neither made nor created, the Son is instead generated by the Father. This generation is the “origin of a living being from a living principal of the same nature.”
But what is the nature of this generation. The Church is very careful about the terms it uses to describe these complex mysteries, which is why the word “creation” is best avoided when addressing the procession of the Son from the Father. In the wake of the Arian heresy, which taught that Jesus Christ was a “creation” of the Father, and thus not equal to the Father, the Church had to be very cautious when using terms of “creation” when referred to the Trinity.
The Arian error was never how the Church understood the clear teaching of Christ and the Apostles. As we read in John 1:1-3 “In the beginning was the Word, and the Word was with God, and the Word was God. He was in the beginning with God; all things were made through him, and without him was not anything made that was made.”
Several things are made clear in these lines. First, the son is co-eternal. He always was. Second, he was not merely “with God,” he “was God.” Third, all things—that is, all created things—were “made through him.” Jesus Christ was not merely equal to God or favored by God, he was of one substance (consubstantial) with God.
It is through knowing and willing that the substance of God became three “persons.” The Father is the “originless origin,” who in turn generates the Son through an act of the intellect. The thought of the Father generates an image of himself that is identical in all ways except Fatherhood. This image is of the same substance with the Father, sharing a single will while remaining diverse in person.
A normal image may be less than the source, much like a copy of an artwork is less than the original. The generated Son, however, is a perfect image of the Father. He is “true God from true God.” In this process of generation, God the Father communicates his substance to God the Son.
It was left to St. Thomas to clarify the exact nature of this generation. In the Summa, he explains two types of procession. One is an outward act of creation, while the other is an “inward procession corresponding to the act remaining within the agent.” This second kind of procession is purely an act of the intellect. If you have a thought, that thought may proceed outwards and become manifested as words or actions, or it may proceed inward as a “conception of the object understood, a conception issuing from our intellectual power and proceeding from our knowledge of that object.”
St. Thomas describes this as an “intelligible emanation.” An idea, even after spoken, continues to exist in the intellect of the speaker. Once you’ve spoken a thought, for example, it’s not erased from your memory. Thus, it becomes possible for an object of the intellect to “proceed” from its origin while remaining part of that origin. The act of intellection is itself a procession.
This divine act of knowing does not result in an accident, as it would for a mortal creature. Since God is not potential but pure act, the result of His knowing is to create a simulacrum of the object being known. What God is knowing is Himself; His own essence, which He obviously comprehends. “The concept of the intellect is a likeness of the object conceived,” write St. Thomas, “and exists in the same nature, because in God the act of understanding and His existence are the same.” This act of self-knowing results in the generation of the Son, identical in all ways except for paternity.
As we move from the Son to the Holy Spirit, we encounter a second kind of procession, which St. Thomas called “spiration.” The procession of the Holy Spirit was a thorny issue even among orthodox Church Fathers and Doctors of the Church, particularly between the Eastern and Western churches. The insistence of the Eastern Churches that the Holy Spirit proceeded only from the Father was one of the instigating factors in the Great Schism.
Yet the Western understanding of the Holy Spirit proceeding from Father and Son was quite common among the Fathers. Even if we assume that the Athanasian Creed was not in fact written by St. Athanasius in the 3rd century, it existed at least by the 5th century, and reflects clearly the thought of St. Athanasius on the subject in asserting that “the Holy Ghost is from the Father and the Son, not made, nor begotten, but proceeding.”
The dual nature of this procession was still under debate almost 1000 years later, causing the Second Council Lyons (1274) to declare that
“the holy Spirit proceeds eternally from the Father and the Son, not as from two principles, but as from one principle; not by two spirations, but by one single spiration. This the holy Roman church, mother and mistress of all the faithful, has till now professed, preached and taught; this she firmly holds, preaches, professes and teaches; this is the unchangeable and true belief of the orthodox fathers and doctors, Latin and Greek alike. But because some, on account of ignorance of the said indisputable truth, have fallen into various errors, we, wishing to close the way to such errors, with the approval of the sacred council, condemn and reprove all who presume to deny that the holy Spirit proceeds eternally from the Father and the Son, or rashly to assert that the holy Spirit proceeds from the Father and the Son as from two principles and not as from one.”
This important (and binding) declaration by the Council clarifies two important points. First, the Holy Spirit proceeds from the Father and the Son. Second, this procession is from a single spiration, proceeding from a single substance. This means that a single act of the will, shared through the consubstantiality of Father and Son, results in the spiration of the Holy Spirit.
The Holy Spirit has its origin not in an act of the intellect, as with the generation of the Son, but in an act of the will. It is, in fact, an act of love between Father and Son. St. Bernard of Clairvaux called the Holy Spirit the kiss of God, saying “If, as is properly understood, the Father is he who kisses, the Son he who is kissed, then it cannot be wrong to see in the kiss the Holy Spirit, for he is the imperturbable peace of the Father and the Son, their unshakable bond, their undivided love, their indivisible unity.”
The Holy Spirit is thus not generated, but proceeds from the love of the Father and the Son through an act of their unified will. Together, they breathe forth the Holy Spirit.
In John 15:26, Jesus says, “But when the Counselor comes, whom I shall send to you from the Father, even the Spirit of truth, who proceeds from the Father, he will bear witness to me.” We see in this quote that 1) the Holy Spirit proceeds from the Father, and yet 2) it is Jesus who will send him. Since Jesus also proceeds from the Father, and has authority to send the Holy Spirit, we may conclude that Jesus is equal in every way to the Holy Spirit.
It’s important to distinguish between generation and spiration in order to truly understand the trinity. If the Son proceeds from the Father and the Holy Spirit proceeds from the Father, then Son and Holy Spirit would be the same. God would have two Sons. This, obviously, is not the case. Each has a distinct origin. The Son is often called the Wisdom of God, proceeding from an act of the intellect. Meanwhile, the Holy Spirit is the love of God—the very breath of God—proceeding from an act of the will: an act of love. In these two great acts we find the two processions which form the eternal Trinity of Father, Son, and Holy Spirit.
 George Joyce, ”The Blessed Trinity,” The Catholic Encyclopedia, Vol. 15 (New York: Robert Appleton Company, 1912), (http://www.newadvent.org/cathen/15047a.htm). (Retrieved 10/6/11)
 Saint Augustine of Hippo; John E. Rotelle; Edmund Hill, The Trinity: The Works of Saint Augustine, (New City Press: Kindle Edition), 284.
 St. Thomas Aquinas, Summa Theologica (complete) (mobi) (MobileReference: Kindle Edition, 2008), Kindle Location 5318.
 St. Augustine, 148.
 Kenneth Baker, S.J., “God Our Father,” Fundamentals of Catholicism Vol. 1 Part 1, Chapter 4 (San Francisco: Ignatius Press, 1995), 30-32.
 St. Augustine, 280.
 Kenneth Baker, S.J. “The Origin of the Holy Spirit,” Fundamentals of Catholicism Vol. 1 Chapter 31 (San Francisco: Ignatius Press, 1995), 92-94
 St. Thomas, Part 1, Q 28.
 The Second Council of Lyons, Papal Encyclicals Online, (http://www.papalencyclicals.net/Councils/ecum14.htm), Constitution II, Section 1. (Retrieved 10/3/11)
 St. Bernard of Clairvaux, “Sermon 8 on the Song of Songs,” (http://www.pathsoflove.com/bernard/songofsongs/sermon08.html). (Retrieved 10/3/11)