Irresistible grace is the Reformed belief that God gives His saving grace only to those whom He has already chosen to be saved. Those people are then drawn to God because He willed it. Those who are not chosen are not given this grace and are therefore condemned in their sins.
This doctrine rejects the possibility of people choosing God of their own free will in any respect. This is because of the strong view of total depravity. In the Reformed view of total depravity, the natural man is so sinful that if he were given the choice to choose God, he would always choose to reject Him. Therefore, the only way God can save men is to choose for them. He then regenerates them before they believe and bend their wills to His own.
This again reinforces how none can boast in their salvation because it was not of any righteousness of their own that they were saved, in accord with the general doctrine of sola gratia.
The primary texts used to support this doctrine are:
Romans 8:30 Moreover whom he did predestinate, them he also called: and whom he called, them he also justified: and whom he justified, them he also glorified.
John 6:37 All that the Father giveth me shall come to me; and him that cometh to me I will in no wise cast out.
Opponents of the doctrine instead point to verses like John 12:32 and Titus 2:11 to instead support the belief that God offers saving grace to all men.