More in the theology series...
1. Is Theology Practical?
2. Why Believe in God?
God as Creator
3. God as Other
4. God as All-Powerful
5. God as All-Knowing
6. God as Eternal
7. God as Spirit
John 4:24 proclaims that "God is spirit, and it is necessary for those who worship him to worship in spirit and truth." As is often the case, the first statement is often lifted out of its context and made to make an abstract theological claim. But as with all words, what were these words doing in context?
In John 4, Jesus is telling a Samaritan woman that neither the temple in Jerusalem or Samaria localize God. God is present in spiritual form, not in physical form. The spirits of human beings are what worship God. To say God is a spirit in this sense is thus to say that he is not visible or tied down to a fixed location. It points us toward the Christian sense that God is everywhere present in the universe, though not often visible.
The New Testament tells us that the Spirit of God spoke through the prophets of the Old Testament (e.g., 2 Pet. 1:21). The Spirit of God reveals truth to his people (1 Cor. 7:40). The Spirit of God empowers his people (Acts 4:8). Most intriguing of all, a few places seem to speak of the Holy Spirit as a distinct person from God the Father (John 14:26; 2 Cor. 13:13).
Of course this is exactly what Christians would officially affirm in the earliest centuries of the church. God the Spirit is a distinct person of the Trinity, of one substance with the Father. The practical implications are that God is with us, wherever we go. He is with us to empower us, as he did to the apostles in Acts. He is with us to give us knowledge of the truth (John 16:13), knowledge of sin, righteousness, and judgment (John 16:8-11).
The Spirit is God's seal of ownership on us (Eph. 1:13-14), a deposit and guarantee of our future existence. Spirit is the stuff of heaven--the biblical authors probably thought literally, although we are best to take it as a picture (spirit was understood in material terms at the time of Christ, just thin material). Romans 8:9 tells us that if someone does not have the Spirit of Christ, they do not belong to him. In the same way, the Holy Spirit is the definitive ingredient of eternal life in Acts (11:18).
The Spirit of God cleanses and purifies our hearts (Acts 15:9). The Spirit sets us apart as God's people, both individually and corporately (2 Thess. 2:13). He "sanctifies" us, cleansing us and setting us apart. He can sanctify God's people thoroughly, in every aspect of our lives (1 Thess. 5:23).
What does it look like to have the Holy Spirit today? Some Christians think you will speak in tongues if you have the Holy Spirit, but the New Testament hardly says anything of this sort and Paul clearly does not think all believers will speak in tongues (1 Cor. 12:30). American Christianity has tended to dramatize receiving the Holy Spirit to where everyone would have some emotional experience patterned off of Acts, but this is hardly our experience of your average person "getting saved." We would normally expect a peace to come but some individuals have blocks that keep them from feeling God's presence.
We must leave it to God rather than formulate some "one shoe fits all" approach. Most people experience peace and a confidence that they are right with God--if they are taught to expect it (remembering that assurance is a fairly recent notion). Some people get a significant feeling of God's presence. Some people get a sense of spiritual power. Some people speak in tongues.
Others must trust in the knowledge that they have put their trust in God, whether they feel anything special or not. At times other believers need to be faith for the troubled. God has given the body of Christ collectively the spiritual authority to forgive sins (e.g., John 20:23). Sometimes we have to exercise this authority on those who, for whatever reason, are troubled by doubt.
There are instances in Scripture where God's Spirit comes on individuals who are not serving him (e.g., Saul) or are even still in the womb (e.g., John the Baptist). So we should not assume that a person must be an adult or a believer to experience the power or presence of God's Spirit. The Spirit can also depart from God's people, as seems to have happened to Israel at one point (cf. Ezek. 10:18).
What does God's Spirit look like against the backdrop of a robust understanding of creation out of nothing? It is fascinating to think that the picture of God as spirit both served to indicate his invisible presence everywhere while also indicating that he transcended the embodied realm. From the standpoint of creation ex nihilo, these are two quite different functions, although they connected well to each other in the worldview of New Testament times.
So in New Testament times, spirit is the stuff from the top of the creation.  If you go straight up, you proceed through layers of sky (=heavens. It's the same word. We only use two different words--sky and heaven--because our view of the world has changed from ancient times). When you get to the highest sky, you get to the highest heaven where beings like angels, "ministering spirits" (Heb. 1:14), were located. From the standpoint of Bible times, this is all part of one uni-verse. 
But with a robust sense of creation out of nothing, we now see more clearly than ever that the essence of God is "outside" this universe. The picture of God as Spirit in Scripture thus comes to refer to two quite different claims, both of which are truth: 1) that the essence of God is "other" than this creation and 2) that God is still nonetheless present everywhere in this world.
The trinitarian formulation of Nicaea in 325, like all revelation, was made within the categories of those to whom God revealed it. There are possibilities in a robust understanding of creation out of nothing for understanding God the Spirit as God in this universe and God the Father primarily as a reference to the essence of God outside this universe, remembering that the biblical language was not trying to explain these sorts of things. They were simply pointing to different things the one God did and they used different names to do so without defining them in anything like a philosophical sense. 
But these sorts of speculations should make us uncomfortable. They are, after all, the stuff of ecumenical councils and the whole of Christendom. It is also, however, a warning to the current trendiness of Trinity-talk right now among theologians. Theology literature right now is rife with what must surely in the end be wild speculation about how the nature of the Trinity impacts us. In the end, such theologians are probably embarrassing themselves beyond comprehension. Most of this speculation, I suspect, will turn out to be impractical, even if it comes in practical garb.
God the Spirit means that God is with us everywhere, even though we cannot see him. God the Spirit means that we have purity and power at our disposal. God the Spirit means that we have direction and knowledge at our disposal. God the Spirit means that we have God at our disposal.
 In Old Testament times, spirit is wind, breath (ruach). The Spirit of God is thus conceptualized as the wind or breath of God.
 The Bible does not seem to have any category for anything being outside the cosmos. It seems to operate with the picture of a three-story universe, with heaven being up and Sheol, the realm of the dead, being down.
 What I am saying here is that "God the Father" and "God the Spirit" is phenomenological language that refers to God as he appeared to his people and was experienced by them in their thought categories, God in the way he functioned with them. Philosophically and theologically, we might at least in theory give a more precise delineation if we had a more precise understanding of their roles. For example, could some functions that were experienced as being functions of "God the Father" have more precisely been functions of God the Spirit?
8. Three in One
9. God as Love
10. God as Just
11. God as Unchanging