Understanding the doctrine of prevenient grace was one of the most valuable studies for me after leaving Calvinism. It provided an answer to one of the simplest arguments I used to make for Calvinism: I would point to Romans 3 and ask “If ‘no one understands, no one seeks for God?’ then how is anyone saved?” My Calvinistic answer, of course, was irresistible grace given to the elect only. Otherwise, I reasoned, either no one would believe, or everyone would believe.
Pre-regenerating grace simply means that the Spirit of God overcomes that inability by a direct work on the heart, a work that is adequate to enable the unregenerate person to understand the truth of the gospel, to desire God, and to exercise saving faith.[…]Scripturally, this concept is intended to express the truth found in passages like John 6:44: “No man can come to me, except the father which hath sent me draw him.” In this light, pre-regenerating grace may be called drawing. Or Acts 16:14: “Lydia…whose heart the Lord opened, that she attended unto the things which were spoken of Paul”: pre-regenerating grace may therefore be called opening the heart. Or John 16:8: “When [the Spirit] is come, he will reprove [or, convict] the world of sin, and of righteousness, and of judgment.” In this sense, pre-regenerating grace may be called conviction–which is simply another form of the word “convincement.”
Arminians argue that not all who are drawn/enabled by the Father to exercise faith and repentance do in fact ultimately choose to do so (i.e., prevenient grace is resistible), though it is equally true that without such drawing/enabling no person would of himself have the desire or ability to come to Christ in faith. Arminians are able to adopt this position precisely because the drawing and enabling of the Father are presented in the Gospel of John as necessary, not sufficient, conditions for coming to faith in Christ.
Christian theology teaches the doctrine of prevenient grace, which briefly stated means this, that before a man can seek God, God must first have sought the man.Before a sinful man can think a right thought of God, there must have been a work of enlightenment done within him; imperfect it may be, but a true work nonetheless, and the secret cause of all desiring and seeking and praying which may follow.We pursue God because, and only because, He has first put an urge within us that spurs us to the pursuit. “No man can come to me,” said our Lord, “except the Father which hath sent me draw him,” and it is by this very prevenient drawing that God takes from us every vestige of credit for the act of coming. The impulse to pursue God originates with God, but the outworking of that impulse is our following hard after Him; and all the time we are pursuing Him we are already in His hand: “Thy right hand upholdeth me.”In this divine “upholding” and human “following” there is no contradiction. All is of God, for as von Hegel teaches, God is always previous. In practice, however, (that is, where God’s previous working meets man’s present response) man must pursue God. On our part there must be positive reciprocation if this secret drawing of God is to eventuate in identifiable experience of the Divine. In the warm language of personal feeling this is stated in the Forty-second Psalm: “As the hart panteth after the water brooks, so panteth my soul after thee, O God. My soul thirsteth for God, for the living God: when shall I come and appear before God?” This is deep calling unto deep, and the longing heart will understand it.
It requires the hearing of the gospel. […T]he Word is the instrument, the means used by the Spirit as a basis for the conviction, the persuasion, the enabling. This observation accords with the concept of the power of the Word of God spoken of everywhere in the Scriptures, as in Hebrews 4:12 for example. Arminius’ view on this is clear when, speaking of the persuasion involved in this pre-regenerating grace, he says, “This is effected by the word of God. But persuasion is effected, externally by the preaching of the word, internally by the operation, or rather the co-operation, of the Holy Spirit, tending to this result, that the word may be understood and apprehended by true faith”.
Differing from Calvinism, those of the Arminian persuasion hold that grace extends to everyone the ability to believe. Just as no support is found to limit the atonement of Christ, so there is no reason to limit saving grace. As Wesley put it, “There is no man, unless he has quenched the Spirit, that is devoid of the grace of God…. it is more properly called preventing grace.”The term prevenient comes from two Latin words that mean “to come before.” Used theologically, it refers to the operation of God’s grace in the heart before one comes to Christ. This preparatory grace is comprehensive, including any movement of man toward God, and involves illuminating divide truth, conviction of sin, call to repentance, and the exercise of saving faith. Yielded to, these gracious impulses increase; when stifled, they tend to diminish. All these promptings of the Spirit imply some awakening of spiritual life, some beginning of deliverance from a heart of stone. 
While this is a difficult question and does involve mystery, here are some of the texts that I have come across which seem to provide some answers:
Jesus makes it clear in Luke 14:26-33 that some will find the terms of discipleship too costly, and therefore reject the invitation (notice that this account follows the parable of the wedding banquet, which in turn follows Jesus’ lament over Jerusalem at the end of chapter 13).
Of course, if Jesus were a Calvinist, He never would have suggested that it was harder for rich persons to be saved by God’s irresistible grace than poor persons. Their wills would be changed immediately and invincibly upon hearing God’s effectual call. It would be no harder for a rich person to be saved by God’s monergistic and irresistible calling than it would be for any other sinner. But the real Jesus was suggesting that their salvation was tied in some measure to their response and commitment to His calling. (Link, p 121).
The book of Hebrews repeatedly warns people who have heard the gospel not to harden their hearts. “Today, if you will hear His voice, do not harden your hearts” (Hebrews 3:7, 15; 4:7). When a person who has been convicted by the Spirit of God hears the gospel, he may choose to resist (Acts 7:51). He thereby insults the Spirit of grace in rejecting Christ (Hebrews 10:29).(Richard Trader, Link)
More resources on prevenient grace:
- Seven Minute Seminary: Chuck Gutenson, What Is Prevenient Grace?
Brian Abasciano, Freed to Believe by God’s Grace in The FACTS of Salvation: A Summary of Arminian Theology/the Biblical Doctrines of Grace
Roger Olson, Prevenient Grace: Why it Matters
Solus Arminus, Granted by the Father: a primer on Prevenient Grace
Kevin Jackson, Prevenient Grace Explained — including a good list of supporting verses.
Arminian Today, How Prevenient Grace Helps Me Sleep
Jesse Owens, Arminius and the Doctrine of Prevenient Grace
- Matt O’Reilly, Common Grace vs Prevenient Grace: What’s the Difference?
W Brian Shelton, Prevenient Grace: God’s Provision for Fallen Humanity — a review by Roger Olson is available here. You can also read the author’s “journey into exploring the biblical foundation for prevenient grace” here.
It is true that the TS does not use this Arminian phrase “prevenience of supernatural grace.” But any concern that Article 2 neglects an emphasis on God’s grace should be assuaged by the following declarations in the Statement:“…no sinner is saved apart from a free response to the Holy Spirit’s drawing through the Gospel.” – Article 2, sentence 4 […]
 For my friends and I (when I was a Calvinist), this understanding of prevenient grace came from Tom Schreiner’s article Does Scripture Teach Prevenient Grace in the Wesleyan Sense? Dr Schreiner states in footnote 21(!) that, “For the purposes of this chapter only Wesley’s later theology of prevenient grace is in view” and points to resources which show “three different understandings of prevenient grace in the Wesleyan tradition” and “two strands of prevenient grace among Wesleyans”, but this is practically hidden from the text of his article where he states only, “In Wesleyan theology there are various conceptions of prevenient grace that we do not need to specify here since, as we shall see, there is common ground within the various positions on the issue that concerns us.” I wish he was clearer to distinguish between the spectrum of Arminian views, as this would have helped to prevent much of the misunderstanding which is so common. (Thomas R Schreiner, Does Scripture Teach Prevenient Grace in the Wesleyan Sense? in Still Sovereign, edited by Thomas R Schreiner and Bruce A Ware)
This post originally appeared at BeyondCalvinism where comments can be made.