Here's my definition again, subject to revision.
For those that God has chosen, there comes a point where God will save them by the working of His Holy Spirit in their lives, bringing them to repentance and faith. This call, done by God's grace, is thus termed “irresistible” as He overcomes all obstacles to faith and saves people.The key point that the Synod of Dordt wanted to make was against "resistible grace."
As I understand it, "resistible grace" depicted God's call to salvation as that of "wooing," "drawing," an "invitation" and a "gift," and thus could be rejected or accepted. That was dependent upon free human choice. (see article 4)
In contrast, the doctrine summarized by "irresistible grace" (in it's more consistent form, as there is disagreement about the sincere offer) rejects the idea of "God invites you" in favor of a divine command to repent and believe the Gospel. It presents God not only as "wooing" but also actively transforming the person. This act, done by the Holy Spirit, is termed irresistible, as it is seen as on the same level as God commanding "Let there be light." (2 Corinthians 4:4-6)
While it certainly is possible for men to resist the Holy Spirit in a sense (Acts 7:51), but there is a point where God says "enough!" and He brings repentance and salvation upon people. Simply put, God transforms their heart so that they can't help but believe.
Irresistible grace is a complement to total depravity. If we accept that man cannot take that step to God, but instead is hostile to God, dead in its sins, a slave to sin, blind, deaf, and possessing a heart of stone, then it must be completely dependent upon God to save him, transform him, raise him from the dead, open his eyes, remove his heart of stone and give him a heart of flesh. For if it were up to man at all, then his depraved nature would reject God completely and turn away from Him. (see my previous post for where all these references are to)
Before I dive into expounding two specific passages, here are some brief references of this actively happening in Scripture.
2 Timothy 2:24-25-
24 And the Lord's servant must not be quarrelsome but kind to everyone, able to teach, patiently enduring evil, 25 correcting his opponents with gentleness. God may perhaps grant them repentance leading to a knowledge of the truth,(God granting repentance)
14 One who heard us was a woman named Lydia, from the city of Thyatira, a seller of purple goods, who was a worshiper of God. The Lord opened her heart to pay attention to what was said by Paul.(Lord opened her heart)
2 Chronicles 30:11-12
11However, some men of Asher, of Manasseh, and of Zebulun humbled themselves and came to Jerusalem. 12The hand of God was also on Judah to give them one heart to do what the king and the princes commanded by the word of the LORD.(Some humbled themselves, and the Lord's hand also was on Judah to give them one heart to obey the Lord)
The two major texts to come: John 6 and Romans 8.