In contrast to Calvinism, the Bible teaches that Christ died for all men, and that God is not willing that any should perish.
1 Tim 4:10 For therefore we both labour and suffer reproach, because we trust in the living God, who is the Saviour of all men, specially of those that believe.
Jesus made the full payment for sins for all men, and any man can be saved. However, the only way to receive this gift is by placing one's faith in Christ.
Rom 9:30 What shall we say then? That the Gentiles, which followed not after righteousness, have attained to righteousness, even the righteousness which is of faith.
31 But Israel, which followed after the law of righteousness, hath not attained to the law of righteousness.
32 Wherefore? Because they sought it not by faith, but as it were by the works of the law. For they stumbled at that stumblingstone;
33 As it is written, Behold, I lay in Sion a stumblingstone and rock of offence: and whosoever believeth on him shall not be ashamed.
Rom 10:1 Brethren, my heart's desire and prayer to God for Israel is, that they might be saved.
2 For I bear them record that they have a zeal of God, but not according to knowledge.
3 For they being ignorant of God's righteousness, and going about to establish their own righteousness, have not submitted themselves unto the righteousness of God.
4 For Christ is the end of the law for righteousness to every one that believeth.
It is not that Jesus did not die for all men. It is the method by which one must take hold of this free gift from God. God only offers it through faith in his son Jesus, a person cannot come any other way. If you try to come by your own works you will fail. If you believe you will be saved because you are the decendent of Abraham or belong to a particular church you will fail. You can only be saved by coming to Jesus Christ in faith.
And it is clear that God does not choose before time who will believe and who will not. For Paul clearly shows that some of these Jews who were cut off because of unbelief can turn and trust Christ.
Rom 11:17 And if some of the branches be broken off, and thou, being a wild olive tree, wert graffed in among them, and with them partakest of the root and fatness of the olive tree;
18 Boast not against the branches. But if thou boast, thou bearest not the root, but the root thee.
19 Thou wilt say then, The branches were broken off, that I might be graffed in.
20 Well; because of unbelief they were broken off, and thou standest by faith. Be not highminded, but fear:
21 For if God spared not the natural branches, take heed lest he also spare not thee.
22 Behold therefore the goodness and severity of God: on them which fell, severity; but toward thee, goodness, if thou continue in his goodness: otherwise thou also shalt be cut off.
23 And they also, if they abide not still in unbelief, shall be graffed in: for God is able to graff them in again.
24 For if thou wert cut out of the olive tree which is wild by nature, and wert graffed contrary to nature into a good olive tree: how much more shall these, which be the natural branches, be graffed into their own olive tree?
Here Paul is speaking of Jews who are lost at this time because of unbelief. They are said to be branches broken off of the olive tree. But their fate has not been decided, Paul shows in vs. 23 that if they turn from unbelief to belief God is able to graff them back into the tree.