Romans 9 is the most commonly used, and I would say misused passage in the entire discussion of the sovereignty of God and the responsibility of man. I can fully understand why Calvinists believe as they believe, and argue as they argue, based on this passage alone. However, I contend, and I hope all Calvinists agree, that Romans 9 was never designed to be taken alone. Not only is it bad practice to read a passage as if it was all the Scripture a man will ever need, but it is also somewhat uncommon for one passage of Scripture to be the only relevant passage of Scripture. It happens, but it's rare.
This is the case with Romans 9. Not only is the subject matter dealt with elsewhere in Scripture, but some of it is a quite explicit quote of previous Scriptures. The famous Romans 9:13, for example, says "As it is written, Jacob have I loved, but Esau have I hated." (Emphasis mine). In other words, this is very explicitly not the only relevant Scripture.
So we come to verses 20-24, and read "Nay but, O man, who art thou that repliest against God? Shall the thing formed say to him that formed it, Why hast thou made me thus? Hath not the potter power over the clay, of the same lump to make one vessel unto honour, and another unto dishonour? What if God, willing to shew his wrath, and to make his power known, endured with much longsuffering the vessels of wrath fitted to destruction: And that he might make known the riches of his glory on the vessels of mercy, which he had afore prepared unto glory, Even us, whom he hath called, not of the Jews only, but also of the Gentiles?"
Taken alone, it's not hard to see how one might draw Calvinist conclusions from this passage. Perhaps, if you are undecided, you might even be compelled to accept Calvinism. But this is not the only passage in which God compares Himself to a potter.
In Jeremiah 18, we read "The word which came to Jeremiah from the Lord, saying, Arise, and go down to the potter's house, and there I will cause thee to hear my words. Then I went down to the potter's house, and, behold, he wrought a work on the wheels. And the vessel that he made of clay was marred in the hand of the potter: so he made it again another vessel, as seemed good to the potter to make it. Then the word of the Lord came to me, saying, O house of Israel, cannot I do with you as this potter? saith the Lord. Behold, as the clay is in the potter's hand, so are ye in mine hand, O house of Israel."
This shows us that God not only has plans, but options. He is fully sovereign over the "clay" (i.e. us, or in this case national Israel). If He wants, He can prosper them. If He wants, He can destroy them. At this point in time, they have even done enough to justify their own destruction. We can imagine it like this: God wanted to make Israel a plate. If they became the plate, excellent, mission accomplished. But if they will not become a plate, very well:
This concept continues in the Jeremiah passage. In verses 7-10, we read "At what instant I shall speak concerning a nation, and concerning a kingdom, to pluck up, and to pull down, and to destroy it; If that nation, against whom I have pronounced, turn from their evil, I will repent of the evil that I thought to do unto them. And at what instant I shall speak concerning a nation, and concerning a kingdom, to build and to plant it; If it do evil in my sight, that it obey not my voice, then I will repent of the good, wherewith I said I would benefit them."
And so we see here that God can actually "repent". He can change His mind. How? Well, when we change our minds, of course. With regard to salvation, God's plan A is our salvation. He "says He will benefit us". What Calvinist will deny this? An inheritance in God's eternal Kingdom, which can be gained through faith alone, is a very good thing, and it is clearly within God's plan. I have shown this in the past with Christopher Hitchens. If Christopher Hitchens had repented, would he be saved? Yes. Would the Gospel have been altered in any way if that had happened? No. The Gospel remains identical regardless of who actually accepts it. Thus, the plan of salvation God put into action is a good which benefits us. Yet, when we "become marred in His hand", He repents of that good, and we become vessels of wrath fitted for destruction, and God merely tolerates us for sake of His glory.
Plan B is that destruction. When we sin, we earn ourselves wrath. When we receive salvation, we become like clay which initially became marred in His hand, and so He plans to "pull down and destroy" us in Hell. But through His grace, He has provided a way for everyone to be saved, and He "repents of the evil He thought to do to us".
To this, a Calvinist might say "then God isn't really sovereign because He can't achieve His goals". To that, I simply say would you like to tell Him that? Notice, all I did was re-state Scripture. I used the identical analogy, I showed where God Himself says "if they do this, I will repent of what I thought". God says our attitude to Him affects His attitude to us. If this makes God weak in your eyes, you are wrong, because you disagree with the Almighty.
The way I like to look at it is this: If we are saved, that's good for God, and good for us. God wins, and we win with Him. If we are condemned, that's good for God, but bad for us. God still wins, but we lose against Him. In any scenario, God wins. His plan is achieved, even if His preferences are not always met. In no less than 3 places in Ezekiel alone, God tells us plainly that He has no pleasure in the death of the wicked (Ezekiel 18:23, 32; 33:11). This further fits in with His claim to not want anyone to perish (2 Peter 3:9), and would rather all men be saved and come to the knowledge of the truth (1 Timothy 2:4). To that end, the cross is sufficient to atone for the sins of the whole world (1 John 2:2). This is the extent of God's purpose: whoever believes in Him will not perish but have everlasting life (John 3:16). Is this the case? Does whoever believes in Christ have everlasting life? Yes. So how can one say that even if the entire world perished in unbelief, God has somehow fallen short?
The potter analogy in Romans 9 fits very well with a non-Calvinist view on the relationship between the sovereignty of God and responsibility of man. In God's sovereignty, He has full control over possible forms for the clay. But He is not a one-track kind of God. He can, and explicitly tells us that He does, repent of both good gifts and terrible punishments. The choice He makes depends on the choice He gives us. The sovereignty of God is in the fact He even provides such options. The responsibility of man is in us choosing the one that's best for both of us.