In 1 Timothy 2:4, we're told that God "...will have all men to be saved, and to come unto the knowledge of the truth." To most people, that's fairly clear. There may well be people who will not be saved, but the Bible at least says God wants them to be. To a Calvinist, however, this verse requires a little editorial work. In order to conform this verse to Calvinism, they argue we should change the verse so it reads "all kinds of men".
Of course, we can settle that debate by simply pointing out that Paul wrote the inspired words of God, whereas John Calvin... well, let's just say I have no reason to believe he held any authority at all, much less the authority to literally re-write scripture. But we can go a little further and point out that 1 Timothy 2:4 is not the only scripture that shows us God doesn't want anyone in Hell.
All throughout scripture, we see God longing for the salvation of those who do not wish to be saved. Contrary to Calvinist assertions, this no more downplays His sovereignty than the fact Jesus was "overwhelmed" by Roman soldiers. In the end, though God desires the salvation of all, we are the losers if we do not accept His salvation. Nevertheless, even in the Old Testament, we see that God does desire we take it. Nowhere is this more clearly visible than in Ezekiel, as in no less than 3 places, God expresses His displeasure at death:
"Have I any pleasure at all that the wicked should die? saith the Lord God: and not that he should return from his ways, and live?" - Ezekiel 18:23
"For I have no pleasure in the death of him that dieth, saith the Lord God: wherefore turn yourselves, and live ye." - Ezekiel 18:32
"Say unto them, As I live, saith the Lord God, I have no pleasure in the death of the wicked; but that the wicked turn from his way and live: turn ye, turn ye from your evil ways; for why will ye die, O house of Israel?" - Ezekiel 33:11
With these verses in mind, it makes sense that, since God takes no pleasure in the death of the wicked, He wants all men to be saved and come to the knowledge of the truth. It makes no sense, however, to say that, since God takes pleasure in the death of the wicked, He only wants all kinds of men to be saved.
Of course, Calvinists do not say this explicitly, but they must believe it implicitly. After all, in their view, the only reason we must change 1 Timothy 2:4 is because no one can resist God's will, and so if God wills all to be saved, all will be saved. But this would mean if even one person goes to Hell, it is because God wills it. In other words, the wicked died because God took pleasure in their death.
The simple fact is we do not serve a contradictory God. His power is limited by His character. Just as He cannot lie (Titus 1:2), He also cannot contradict Himself. Therefore, in order to create free willed beings, which are absolutely necessary in order for evil to even be a possibility (for God cannot be tempted nor tempt anyone, James 1:13-14), God must allow those He wishes to save to instead be condemned.
Because God takes no pleasure in the death of the wicked, desiring repentance instead, we must naturally reject Calvinism, as Calvinism must contend that if the wicked die, it is because God willed it. The Bible is beyond clear on this topic: God wills that all men be saved, Christ died for the salvation of all, and so it is up to all men to accept the gift of salvation. If they refuse it, it's on them.