To expand a little of DJClayworth’s answer, it is worth going into exactly what “spirit” means and how it applies to the Triune God.

The meaning of the term “spirit”

In short “spirit” (in the Greek, both classical and New Testament: pneuma) brings together two rather disparate concepts, one from the Old Testament, and one from the world of classical Greek philosophy.

In the Old Testament, pneuma is the translation of the Hebrew term ruah, which literally means “breath” or even “wind.” It can refer to God Himself, as in Genesis 1:2:

And the Spirit of God was hovering over the face of the waters.

to the very life of a man, as in Job 27:3:

as long as my breath is in me, and the spirit of God is in my nostrils,

sometimes to life in general (specifically animal life, which breathes), as in Psalm 104:29:

When you hide your face, they [the creatures of the sea and the ships that sail on it—and presumably their human occupants, too] are terrified; when you take away their breath, they die and return to the dust.

and finally to a special inspiration or indwelling of God, as in Judges 6:34:

But the Spirit of the Lord clothed Gideon, and he sounded the trumpet, and the Abiezrites were called out to follow him.

When the Israelites came into contact with Greek culture, the term pneuma began to take on a more technical, philosophical meaning; namely, that which is beyond the purely material world and does not depend on matter for its existence.

In this technical sense, man is a spirit (albeit an incarnate spirit), because, although he cannot exist fittingly without a body, still the death of his body does not cause the dissolution of his being. (See Catechism of the Catholic Church [CCC] no. 362) Angels are also spirits—in this case, pure spirits that do not depend on a body at all. (See CCC 328.) And all the more is God pure spirit. (Although it is not Magiterium, a good treatment of this topic is Summa theologia [S.Th.], Ia. q. 3, a. 1.)

It is in this context that the early Church called the Third Person of the Trinity the Holy Spirit. Jesus had referred to the Third Person as the “Spirit of Truth” and the “Comfortor” or “Advocate” (Paraclete), and, in fact, He was already revealed as the “Holy Spirit” at the Annunciation in Luke 1:35.

However, none of this takes away from God’s (and hence the other Persons’) essentially non-corporeal—that is, spiritual—nature.

Regarding the Holy Trinity

The Church teaches that God is utterly one and simple (S.Th. Ia., q. 3, a. 7; also the Nicene-Constantinopolitan creed: “I believe in one God….”) In God there is no division, composition, admixture, or change of any kind whatsoever.

Hence, each of the Persons are fully and completely God. None of them is a “part” of God, but each of them is (if we can say this) “all” of God. In technical terms, there is no real distinction between God’s substance (also called essence or nature) and each of the Persons. The Father is God, completely and entirely; the Son is God, completely and entirely; and the Spirit is God, completely and entirely.

The Persons, however, are really distinct from one another, and what distinguishes them is their relationship to one another (and they are distinguishible only in this respect). The Father begets the Son; the Son is begotten of the Father; the Father and the Son (or, in the Eastern conception, the Father through the Son) spirate the Holy Spirit; and the Holy Spirit proceeds from the Father and the Son (or—in the Eastern formulation—the Spirit has his ultimate origin in the Father, but receives the Divine Essence through the Son). (See CCC 254-255.)

The Persons of the Trinity are not merely “manifestations”

It is not correct, therefore, to conceive of the Persons as mere manifestations of God—as if the Persons were mere appearances, but not really distinct from each other. That would be tantamount to the heresy of Modalism. As the Catechism explains,

“Father”, “Son”, “Holy Spirit” are not simply names designating modalities of the divine being, for they are really distinct from one another (CCC 254, emphasis added).

It is true that the Father sent two of the Persons into the world in a special way: the Son, in the Incarnation; and the Holy Spirit, at Pentecost. These are the Trinitarian Missions (from the Latin mitto, to send; see CCC 275-260). However, the Persons have existed for all eternity.

All of the Persons are “spirit”

Since each of the Persons is identical to the Divine Essence (i.e., is fully and completely God), it follows that each of them has all of the attributes proper to God, which includes being “spirit”—that is, being non-corporeal. Hence not only the Holy Spirit, but all of the Persons are pure spirit.

Regarding the names of the Persons: why “Holy Spirit” if all the Persons are “spirit”?

However, it is fitting that the Third Person of the Trinity should be given the name of Spirit by attribution. Some of God’s attributes are particularly apt for one of the Persons of the Trinity. For example, Creator is especially apt as a name for the Father, because of the obvious similarity between the Father’s begetting of the Son and His creation of the universe. However, in reality, both Son and Spirit are just as much Creator as the Father.

The name “Holy Spirit” comes about in a similar way. The name “Father” is especially fitting for the First Person of the Trinity because of the similarity between human fatherhood and the Fatherhood of the Father. Likewise, the name “Son” is fitting for the Second Person, because of the similarity between His being begotten and the begetting of human offspring. However, the Procession of the Holy Spirit bears no such similarity to human reality; for us, the world of “spirit” is mystrious and hidden, in much the same way as the reality of the Procession of the Holy Spirit is hidden from our understanding. Hence, the name “Holy Spirt” is fitting for the Third Person. (See S.Th. Ia., q. 36, a. 1.)