I'd like to write a more thorough summary of the book, but that's going to take more time than I presently have. So, in the interests of sharing at least some of what I've been learning from Owen, here's a shorter post on just one part of Owen's book.
In chapter six, having already discussed the matter of prayer, Owen takes up the role of the Spirit in the manner of prayer and outlines four specific things the Spirit does in helping us to pray as we should.
And, in typical Owenian (and Puritan) fashion, there are also a few sub-points along the way!
In what follows, I've paraphrased Owen, except when direct quotes are used. All the quotes are from volume 4 of Owen's Works. 
The Spirit works on our wills and affections.
He not only enables us to pray, he also gives us affections suited to the things we pray about. "And in this work of the Spirit lies the fountain of that inexpressible fervency and delight, of those enlarged labourings of mind and desires, which are in the prayers of believers, especially when they are under the power of more than ordinary influences from him." (p. 288)
Owen's main Scriptural support for this comes from Romans 8:26-27, where Paul writes about the Spirit's intercession for us with groanings that cannot be uttered. Owen understands this to be "nothing but his working in us and acting by us that frame of heart and those fervent, labouring desires, which are so expressed, and these with such depth of intension and labouring of mind as cannot be uttered." (p. 288). I'm not sure this is the only thing meant by the Spirit's intercession, but surely Owen's general point stands.
The Spirit works in the soul an "inward labouring of heart and spirit," that is, a "holy, supernatural desire and endeavor." (p. 288)
This point seems somewhat coextensive with the first. And, in fact, Owen's following exposition is primarily focused on distinguishing the intercessory work of the Spirit in Romans 8:26-27 from the intercessory work of Christ in Romans 8:34.
But I think Owen here intends to describe the Spirit's work in giving us earnestness in prayer, whereas the first point had more to do with the affections themselves. Think of point #1 as having to do with the kind of affections we need, and point #2 with their degree.
The Spirit gives the believer "a delight in God as the object of prayer." (p. 290)
Owen now has in mind a very specific and necessary kind of affection, which he further characterizes as a "a filial, holy delight in God...such as children have in their parents in their most affection addresses unto them" (p. 291). ("Filial," by the way, is an obscure word that means "befitting a son or daughter.")
This is what Paul means when he says that the Spirit causes us to cry out "Abba, Father."
This kind of delight is important, according to Owen, because "without it ordinarily the duty of prayer is not accepted with God, and is a barren, burdensome task unto them by whom it is performed" (p. 290-291).
Owen then discusses three things included in this delight.
"A sight or prospect of God as on a throne of grace."
See Hebrews 4:16
Owen further clarifies that this prospect is by "spiritual illumination" or faith, and that it the throne of grace is the holy place which we enter into with boldness through the blood of Jesus. See Hebrews 10:19. "God, therefore, on a throne of grace is God as in a readiness through Jesus Christ to dispense grace and mercy to suppliant sinners."
This is Owen at his gospel-centered best. He is showing us how the Triune persons of the Godhead work together in relation to prayer. The Spirit gives us access to the Father through the Son. See Ephesians 2:18. In Owen's words, "it is the work of the Spirit, who alone, in and through Christ, revealeth God unto us, and enableth us to discern him in a due manner . . . All the acquaintance which we have with God, in a way of grace, is from the revelation made in us by his Spirit" (p. 292).
"Unto this delight is required a sense of God's relation unto us as a Father."
See 2 Corinthians 6:18, Ephesians 2:18, and Romans 8:16
"There is nothing more essential" to the duty of prayer than this: that "we address ourselves unto God under the notion of a Father; that is, the Father of our Lord Jesus Christ, and in him our Father also. Without this we cannot have that holy delight in this duty which is required in us, and the want whereof ordinarily ruins our designs in it." In other words, without the faith-fueled delight of a child in God as our Father, prayer is ruined.
But the only way to get this kind of child-like delight in the Father is from the Holy Spirit.
This delight is also characterized by boldness.
See Hebrews 10:19, 22
This boldness includes both (a) "freedom of speech" (2 Corinthians 3:17) and (b) "confidence of acceptance."
- "Freedom of speech" is the ability of the heart to "express all its concerns unto God as a child unto its father."
- And by "confidence of acceptance" Owen means not the assurance that we'll get every single thing we ask for, but rather the "holy persuasion that God is well pleased" with our prayers and accepts us when we come to his throne.
The Spirit keeps believers focused on Jesus Christ "as the only way and means of acceptance with God."
See John 16:16, Ephesians 2:18, Romans 5:2, and Galatians 4:6
The Spirit of God is "the Spirit of the Son." He has been sent to glorify Christ in our hearts. And it is because of his work in our hearts that we call out to God as Father. "And hereof believers have a refreshing experience in themselves; nor doth any thing leave a better savour or relish in their souls than when they have had their hearts and minds kept close, in the exercise of faith, on Christ the mediator in their prayers" (p. 296).
Summary
So, how does the Spirit work in our prayers?
- He inclines our wills and stirs our affections towards God
- He gives us earnestness in seeking God
- He gives us delight in God as our Father and boldness to approach his throne of grace
- He keeps us focused on Christ as the sole means of approaching God
So, if you want a diagnostic tool for your prayer life, try asking these questions:
- Am I relying on the Spirit to incline my heart to God?
- Am I trusting in the Spirit to make me earnest in prayer?
- Am I approaching God's gracious throne with the free and delightful boldness of an accepted child?
- Am I trusting in Christ alone to give me access to God?
Summary by Brian Hedges
Prefatory note.
Analysis.
Preface to the reader.
Chapter I. The use of prayer, and the work of the Holy Spirit in it.
Chapter II. Zec 12.10 explained and vindicated.
Chapter III. Gal 4.6 explained and vindicated.
Chapter IV. The nature of prayer — Rom 8.26 explained and vindicated.
Chapter V. The work of the Holy Spirit as to the matter of prayer.
Chapter VI. What the due manner of prayer consists in.
Chapter VIII. The duty of external prayer by virtue of a spiritual gift, explained and vindicated.
Chapter IX. Duties inferred from the preceding discourse.
Chapter X. Of mental prayer as pretended to by some in the Church of Rome.
Chapter XI. Spiritual ability in prayer, in opposition to prescribed forms of prayer
by John Owen