Irresistible grace is a doctrine known primarily within Reformed Theology (Calvinism) which states that the grace with which God saves an individual cannot be successfully resisted by the sinner when he becomes saved. This does not mean that unbelievers cannot generally resist the grace of God, for they surely do. It means that when it comes time for a person to become saved, the saving grace of God upon them cannot be successfully resisted. It is effectual because God's grace overcomes the rebellion of the unregenerate.
Within Reformed Theology irresistible grace is also associated with election and predestination. Election is God's choosing people to be saved, and predestination is his work of manifesting that salvation within them. It is in the preaching of the gospel that the effectual call of God upon the sinner is manifested and the sinner becomes saved. The sinner cannot and will not successfully resist the sovereign work of God to regenerate him when it is the moment of salvation that is wrought by God's power.
However, this grace is not a forceful violation of the will of the recipient. Grace, after all, is the unmerited favor of God upon individuals, but God works upon the person through conviction, the scriptures, and the preaching of the gospel that brings the person to salvation. An illustration can be found with a light bulb. Whenever electricity is present, light is also present. But the light is there because electricity is there. It is not true that electricity there because the light is there. Likewise, when regeneration is present, belief is also present. Belief is the result of regeneration. So, the grace of God upon the sinner always accomplishes what God intends it to accomplish, and the result is regeneration which in turn results in belief. In this way, it is irresistible.