The basic goal of this article is to give a brief description of the function of the Spirit in the Christian life according to the Galatian letter, and then to outline what Paul means when he calls Christians to "walk by/in the Spirit" in Galatians 5:16.
Without simplifying too much, Paul's overarching argument in Galatians is that both Jewish Christians and Gentile Christians receive the promises originally given to Abraham, not through any works of the Law, but through faith in the faithfulness of Jesus Christ. To depart from this simple gospel in even the slightest way by adding even one work of the law to the equation (like circumcision, or washings, or feast days) is to be severed from Christ according to Paul (5:4).
Now, it seems to me that for Paul the Spirit is central to the reception of the promises offered in the gospel to both Jew and Gentile. In fact, upon my reading of Galatians, it seems as if the Spirit is the One in and through whom we actually have the promises. Or, perhaps I could say it in another way, it seems as if the Spirit just is "the promise" from which all the other benefits of salvation flow. Notice how Paul argues in this way throughout the letter: It is the Spirit we recieve through faith (3:2); it is by the Spirit we begin the Christian life and are perfected in it (3:3); it is the Spirit who is both the gift from the Father and the One by whom we have the miraculous sign gifts of the early Christian community (3:5); the "blessing" proclaimed to Abraham in the OT that all the nations would receive seems to be the reception "of the promised Spirit through faith" (3:14, 22); it is by the Spirit that we clothe ourselves with Christ, presuming that being "baptized into Christ" refers to Christ's baptizing us with His Spirit (Mk. 1:8; Gal. 3:27); it is by the Spirit that we, from the heart, are enabled to cry out to God as our Father, and thus become sons and heirs (4:5-7); this adoption is a miraculous birth by the Spirit, just like Isaac's (4:29); it is through the Spirit that we wait for the eschatological hope of righteousness (5:5); it is in or by the Spirit that we live lives of Christian holiness, putting to death the deeds of the flesh and producing new-creational fruit (5:16-26); and finally, it is from the Spirit that we reap eternal life (6:8). Thus, to sum up the general place of the Spirit in the Christian life, it is by the Spirit that we begin the Christian life, are perfected in it, and receive the eschatological gifts of righteousness and eternal life.
With all of that said, I want to briefly answer what I think Paul means when he instructs us to "walk in/by the Spirit" (5:16). This is a question which was left unanswered for me for quite sometime, as many of the explanations I received from fellow Christians or commentators seemed rather vague or mystical, and thus helped to prevent this text from having any practical import in my life as Paul certainly intended it to have. So, in as basic of terms as possible, it seems to me that what Paul means by "walk by the Spirit" is just that Christians should, by the indwelling power and leading of the Spirit, pursue those godly characteristics outlined in Galatians 5:22-23, and avoid those fleshly characteristics outlined by Paul in 5:19-21. That's it. It's that simple. There is no mysterious process by which you and I have to seek to enter into the "realm of the Spirit," or any kind of higher state. It's not as though we can be Christians and yet somehow be cut off from the Spirit and not led by Him. Rather, all who are sons of God by adoption are led by the Spirit of God to put to death the deeds of the flesh (Rom. 8:9-17). It's when we recognize this, that the Christian life becomes much more simple and we can be freed from a kind of anxiety which follows a lack of clarity in our covenantal responsibilities. We don't have to first seek out the Spirit and "get into Him" before we can produce new-creational, Christ-magnifying fruit. Rather, God has graciously sought us out and put His Spirit "into us" to lead us and to give us life. Thus, all you and I have to do is, by a constant faith-filled reliance upon the indwelling Spirit, prayerfully pursue love, joy, peace, patience, goodness, faithfulness, gentleness and self-control.