The Catholic Perspective
I first ran across the word "spiration" in the discussion of the Holy Spirit's relation to the Trinity, found in Aquinas' Summa Theologica (First Part, Question 27). Aquinas has just finished talking first about God as an individual being, and then has begun discussing what it means for God to be a Trinity.
He begins by talking about language in the Bible which seems to indicate that the Son and Spirit, whom we recognize as God, have proceeded from God (e.g. John 8:42 "I came from God"). The procession of Son from Father, Aquinas concludes, can be thought of as analogous to the way that a thought or a word proceeds from the mind, and yet is in a sense part of the mind. He considers it to be "generation", in the sense that (just as in sexual generation) what is generated bears a fundamental similarity in essence to what generates.
What about the Holy Spirit? he then asks. First, he establishes that this procession of Son from Father is not the only procession in God:
Procession exists in God, only according to an action which does not tend to anything external, but remains in the agent itself. Such an action in an intellectual nature is that of the intellect, and of the will. The procession of the Word is by way of an intelligible operation. The operation of the will within ourselves involves also another procession, that of love, whereby the object loved is in the lover; as, by the conception of the word, the object spoken of or understood is in the intelligent agent. Hence, besides the procession of the Word in God, there exists in Him another procession called the procession of love.
(Summa Theologica, First Part, Question 27, Article 3)
That is, "procession in God" comes from God, and terminates in God. The Son is one such procession, just as our words and thoughts proceed from us, and terminate in us. But there is another type of procession, he says. Our intellect produces our thoughts; but our desire—our wants, our ability to choose—produces love. Love also proceeds from us, and terminates in us; though it may influence our actions. In the same way, Aquinas concludes, the Holy Spirit proceeds from God, and terminates in God—from the love the Father and Son have for each other.
So, the last part comes, and the one that directly addresses your question: what should we call this kind of procession? Is it generation, as the Father generates the Son? Aquinas says No:
We must consider that the intellect and the will differ in this respect, that the intellect is made actual by the object understood residing according to its own likeness in the intellect; whereas the will is made actual, not by any similitude of the object willed within it, but by its having a certain inclination to the thing willed. Thus the procession of the intellect is by way of similitude, and is called generation, because every generator begets its own like; whereas the procession of the will is not by way of similitude, but rather by way of impulse and movement towards an object.
So what proceeds in God by way of love, does not proceed as begotten, or as son, but proceeds rather as spirit; which name expresses a certain vital movement and impulse, accordingly as anyone is described as moved or impelled by love to perform an action.
As in creatures generation is the only principle of communication of nature, procession in God has no proper or special name, except that of generation. Hence the procession which is not generation has remained without a special name; but it can be called spiration, as it is the procession of the Spirit.
That is: what's "real" about thoughts, the products of the intellect, are their similarities to what's being thought of—we create a mental image of an apple, and we know it to be a mental image of an apple because it's like an apple. It's this similarity, as I mentioned, that leads Aquinas to call the process "generation". But what's real about the products of the will, Aquinas points out, is not that they're similar to their focus, but that they lead toward their focus. The difference between a mental image of an apple (the product of the intellect) and a desire for apples is that one produces something like an apple, whereas one produces an impulse to go get an apple.
For this reason, Aquinas concludes, it's not appropriate to call this sort of "procession" generation. But, he says, we really don't have any other word for that kind of relationship; so we make up a name, and call it spiration.
A BRIEF NOTE ON THE ORTHODOX VIEW
The Eastern Orthodox Churches, generally speaking, take much less of a structured philosophical approach to details of Church teaching. In particular, they don't worry so much about exactly how it is that the Spirit comes from the Father. Instead, they focus more simply on the Scriptural texts that talk about the Holy Spirit, considering the exact philosophical framework much less important than the personal encounter with the Spirit. Thus, though they developed the concept of "spiration" (which they call προβολή, probolē: "a throwing out"), there's less emphasis in Orthodox theological tradition on philosophical analysis of what this process means and how it takes place, and more emphasis on the sacred encounter with the Spirit in and through the Sacraments.