Irresistible Grace teaches that God intends to save a person, the grace given to him or her at that time will most definitely result in his or her salvation. It does not teach that sinners can never resist God's grace, for we all do that all the time, but that the grace which intends to save will save.
Historically, this is grouped together in the Canons of Dordt in discussions of Total Depravity, in the Third ad Fourth Heads of Doctrine. This was done as both of them had to do with the will of Man. Concerning Total Depravity, did depravity extend to the human will? Concerning grace, is saving grace resistible by the will of Man? The answer is that the grace which saves will effect its own work on the human will. Or to cite from the Canons:
In this way, therefore, faith is a gift of God, not in the sense that it is offered by God for man to choose, but that it is in actual fact bestowed on man, breathed and infused into him. Nor is it a gift in the sense that God bestows only the potential to believe, but then awaits assent--the act of believing--from man's choice; rather, it is a gift in the sense that he who works both willing and acting and, indeed, works all things in all people produces in man both the will to believe and the belief itself. (CD 3/4: 14).
Biblically, it is God the Spirit that brings people to spiritual life (Ezek. 37:1-10; Jn. 3:5-6). The ones who come to Jesus in faith are drawn by the Father through the Spirit (Jn. 6:44). Thus, the Spirit brings life to believers effecting the drawing of the elect to faith in Christ.
Perseverance of the Saints
Perseverance of the Saints teaches that those who are truly saved by Christ will persevere in the faith and will not lose their faith. This was opposed by the Remonstrants and even John Wesley, who believed that perseverance is conditional upon faith. Now, both sides believe that Christians ought to have faith. The question is not whether faith is necessary for salvation, neither is it whether professing believes can fall away, but rather who is responsible for upholding personal faith. (Concerning the first, everyone agrees that faith is necessary for salvation. Concerning the second, Scripture itself states that those who fall away were never true believers in the first place (1 Jn. 2:19).) Thus on the issue of real difference, Calvinists say that it is God who will sustain the faith of someone who is truly saved, while Arminians will put the onus on the believer to sustain his or her personal faith.
Biblically, in John 6:37b, Jesus implies that He is keeping His sheep safe, and in verse 44b, He promises the person drawn by the Father will be present "on the last day." In John 10:28, Jesus clearly states that no one can take His sheep out of His hand. "No one" means that we ourselves cannot take ourselves out of the saving hand of Christ. The reason why the saints will persevere is not because they are such great and good individuals, but because the God they believe in IS great and powerful to save. Thus, we can say:
What then shall we say to these things? If God is for us, who can be against us? He who did not spare his own Son but gave him up for us all, how will he not also with him graciously give us all things? Who shall bring any charge against God's elect? It is God who justifies. Who is to condemn? Christ Jesus is the one who died—more than that, who was raised—who is at the right hand of God, who indeed is interceding for us. Who shall separate us from the love of Christ? Shall tribulation, or distress, or persecution, or famine, or nakedness, or danger, or sword? (Rom. 8:31-35)
For I am sure that neither death nor life, nor angels nor rulers, nor things present nor things to come, nor powers, nor height nor depth, nor anything else in all creation, will be able to separate us from the love of God in Christ Jesus our Lord (Rom. 8:38-39)
No one is able to separate us from Christ and His love, and that includes us ourselves.