Steve’s explanation touches on a part of the problem in grasping the concept of “the procession of the Holy Spirit,” but there is a bit more to it than that.
Two distinct things are meant in Trinitarian theology by the term, and it’s important to distinguish between them: first, the ontological origin, and second, the sending, of the Holy Spirit. Both are, of course, capable of attack on the grounds that numerous people have already addressed in posts above. However, let me try to explain it on the presumption of Trinitarian theology – in other words, for the sake of grasping what is and is not said by the two viewpoints, let’s take as a “given” that they are talking about something real within their frame of reference. Certainly the place of Gitchi Manitou within the Weltanschaaung of philosophical Hinduism would be an interesting question, but of far more relevance to everyday discussions would be what the Hurons themselves meant by the name.
According to orthodox Trinitarianism, God consists of three discrete Persons united in one Godhead: neither three roles played by one God, nor three distinct Gods somehow mystically united – but one God manifesting himself as three persons. All three persons are equally God, equally not each other, equally eternal. There was never a time, on this approach, when God was one Person.
And yet, the Father is ontologically prior to the Son and Holy Spirit. He, the ultimate Ground of All Being, is what they owe their existence to. This has a nasty taste of “All persons of the godhead are created equal, but some are more equal than others.” However, it’s the way in which it’s understood. To construct an analogy: imagine placing a stack of three books on a table. There was never a time when one book was on the table but not the other two. Yet unquestionably the bottom book holds up the other two. Its presence is responsible for their position above the surface of the table. Likewise: the Son is as eternal as the Father, and the Holy Spirit as either of them. Both had roles in the Creation story: the Son as the Word of God through whom all things were made, the Spirit as hovering over the waters. Yet in a real sense, the Father is the source from which the Son was spiritually begotten, and the Holy Spirit proceeded, as of a breath exhaled. (Remember that only in English is it possible to verbally distinguish between spirit, breath, and wind – Greek, Latin, and Hebrew use the same words: *ruach, pneuma, spiritus *for all three.)
Hence the Orthodox insistence that the Holy Spirit proceeds from the Father. However, Catholic theology has an interesting twist to this that I do not fully understand, and would hope that Sarah, tom~ or Bricker might be able to better expound. In their conception, the Spirit is generated as a consequence of the loving interaction of Father and Son.
It might be worth self-hijacking here for a second to address the idea of “begetting” in a divine context. And we need to do a little anthropology to get there. Remember that anyone who has ever been, or been intimate with, a male over the age of puberty is quite well aware of the ejaculation of semen. The ovum, on the other hand, is barely noticeable to the naked eye, and only if you are already aware of its existence and looking for it. In consequence, the view of procreation that prevailed in many primitive cultures is one where the male contributed the “seed” (semen) that grew into the child, the female being relegated to the role of place of fertile growth. More or less, “if you shoot your semen inside her, it will grow into a child in her fertile interior – but it’s your child; you’re the inseminating force that produces it.” (This of course has a lot of sexist results, which are worth going into elsewhere, but for now, grant the naive logic of the belief.)
A man might be a potter, carpenter, blacksmith. He takes stuff other than himself: clay, wood, iron, and shapes it into things other than himself. In contrast, in sexual intercourse, he takes that which is a part of himself, his semen, and causes the woman to produce something of the same sort as himself: his son or daughter. He makes the products of his trade; he begets his children. They are derived from and a part of him in a way that a crock, table, or sword is not.
It is crude and anthropomorphic to conceptualize God as shooting semen into Mary to produce Jesus. But, on the other hand, human beings are made by God, out, according to the myth, of the clay of the Earth, Adam from adamah, Edom (Esau) the Ruddy from edom, red clay. In contradistinction to this, God the Father’s causing of the Son is the creation of something of the same kind as Himself, a second Person of the godhead. The analogy to begetting as opposed to making is painfully obvious.
Okay, back to the Holy Spirit. In its ontological nature, it is the exhalation of God, the spirit/breath coming forth from him and giving life to all things. But in a quite different sense, the Spirit serves as “God acting within one” to believers through its indwelling starting at Pentecost (see Acts 2 for details). And this coming of the Spirit on believers is called for/prophesied at length by Jesus just before his own Crucifixion: “I will ask of the Father, and He will give you another Advocate to be with you forever: the Spirit of Truth, whom the world can never receive since it neither sees nor knows Him. But you know Him, because He is wiith you, He is in you.” (John 14:16-17)
It is very easy to confuse the ontological origin of the Holy Spirit as “proceeding from” the Father in that bizarre ontologically-prior-and-posterior concept, with the sending of the Spirit at Pentecost, done by the Father at Jesus (the Incarnate Son)'s behest.
Hence Eastern Orthodox and Western Catholicism and Protestantism have butted heads about this concept for a thousand years. There was, a few years ago, an agreed statement on the subject produced by a Catholic-Orthodox study group, though whether it will have any lasting effect on the dispute remains to be seen.
Now, everyone is of course free to reject the entire conceptual framework in which this distinguo is founded. But it would be useful if they understood what it is that’s being said and what is not being said, in the process.