Let's take a look at Romans 9 now.
The context of Romans 9 is given in the first few verses.
"I am speaking the truth in Christ—I am not lying; my conscience bears me witness in the Holy Spirit— that I have great sorrow and unceasing anguish in my heart. For I could wish that I myself were accursed and cut off from Christ for the sake of my brothers, my kinsmen according to the flesh. They are Israelites, and to them belong the adoption, the glory, the covenants, the giving of the law, the worship, and the promises. To them belong the patriarchs, and from their race, according to the flesh, is the Christ, who is God over all, blessed forever. Amen." Romans 9:1-5 (ESV)
To summarize, he brings up the anguish he has as a result of his fellow Israelities (ethnic Israel) is cut off from Christ, although they have all these blessings (patriarchs, Christ's line of descendants, adoption, glory, covenants, the law, worship and promises etc.).
His next statement then reveals what we are dealing with.
"But it is not as though the word of God has failed." Romans 9:6a (ESV)
Paul is defending God's faithfulness to his word, on account of Israel's exclusion and unbelief. His reasoning?
"For not all who are descended from Israel belong to Israel," Romans 9:6b (ESV)
The Calvinist will then reason that this is because it is those of the faith who are truly Israel, and therefore ethnic Israel is of no bearing with regards to the Covenant. This can be supported, as you could just go to Romans 4, where Paul talks about this. However, that is not Paul's reasoning here.
"and not all are children of Abraham because they are his offspring, but “Through Isaac shall your offspring be named.” This means that it is not the children of the flesh who are the children of God, but the children of the promise are counted as offspring. For this is what the promise said: “About this time next year I will return, and Sarah shall have a son.” And not only so, but also when Rebekah had conceived children by one man, our forefather Isaac, though they were not yet born and had done nothing either good or bad—in order that God’s purpose of election might continue, not because of works but because of him who calls— she was told, “The older will serve the younger.” As it is written, “Jacob I loved, but Esau I hated.”" Romans 9:7-13 (ESV)
His point here is that it was God's choice. He alludes to the the time where Isaac had two sons, and God chose one and not the other, which was not on the basis of any works performed by them. This seems like unconditional election right? It does, if we lose sight of the context. Which we must realize is about Paul's justification for God's faithfulness in light of Israel's rejection of the Messiah and exclusion from the New Covenant.
That like the time with Jacob and Esau, God has made a sovereign choice. What that choice is will be seen shortly.
"What shall we say then? Is there injustice on God’s part? By no means! For he says to Moses, “I will have mercy on whom I have mercy, and I will have compassion on whom I have compassion.” So then it depends not on human will or exertion, but on God, who has mercy. For the Scripture says to Pharaoh, “For this very purpose I have raised you up, that I might show my power in you, and that my name might be proclaimed in all the earth.”So then he has mercy on whomever he wills, and he hardens whomever he wills." Romans 9:14-18 (ESV)
Paul continues with raising another issue, is there injustice with God's right to sovereignly choose? Paul's answer is no. For God reserves the right to have mercy on whomever he will. That God's Covenant faithfulness and decision on who to enter into is his prerogative. There is more I could say here, but I think the rest of this chapter is explained in concluding statements. Even though this chapter should be seen as part of a unit, Romans 9-11.
"You will say to me then, “Why does he still find fault? For who can resist his will?” But who are you, O man, to answer back to God? Will what is molded say to its molder, “Why have you made me like this?” Has the potter no right over the clay, to make out of the same lump one vessel for honorable use and another for dishonorable use?" Romans 9:19-21 (ESV)
Paul uses the example from Jeremiah, where Israel is the clay and God is the potter and he has a right to do with the clay as he pleases. That he could make from that same clump of clay, Israel, vessels for honorable use and vessels for dishonorable use. That he has taken Israel, and used the obedient and disobedient for his sovereign purposes.
"What if God, although willing to demonstrate His wrath and to make His power known, endured with much patience vessels of wrath prepared for destruction?" Romans 9:22 (NASB)
This is Paul's summary, which I will now summarize. That God was willing to demonstrate his wrath, but instead he chose to have patience on disobedient Israel. Why?
"And He did so to make known the riches of His glory upon vessels of mercy, which He prepared beforehand for glory, even us, whom He also called, not from among Jews only, but also from among Gentiles." Romans 9:23-24 (NASB)
That God has allowed their disobedience to continue, in order to have mercy on more and more people! Which he states so brilliantly in his final summation in Romans 11.
"For just as you once were disobedient to God, but now have been shown mercy because of their disobedience, so these also now have been disobedient, that because of the mercy shown to you they also may now be shown mercy. For God has shut up all in disobedience so that He may show mercy to all." Romans 11:30-32 (NASB)
That unlike the Calvinist's understanding of this text, that it isn't about God's sovereign right to ordain the salvation and reprobation of every individual, but rather a defense of his faithfulness and pointing to God's great plan for mercy to extend to all! That it isn't about limiting God's grace, but expanding it to the greatest amount possible, and that he would use things such as Israel's disobedience and the Gentiles disobedience in order to have mercy on more and more.
Paul then follows up with quoting Hosea to demonstrate that this is fulfilling God's plan all along.
"As indeed he says in Hosea,
“Those who were not my people I will call ‘my people,’
and her who was not beloved I will call ‘beloved.’”
“And in the very place where it was said to them, ‘You are not my people,’
there they will be called ‘sons of the living God.’” Romans 9:25-26 (ESV)
This, like many other texts is about the inclusion of the Gentiles in the New Covenant, and God's present purposes for Israel in their disbelief.