Calvin, among his other points includes the point that we are Unconditionally Elected and chosen by God alone and nothing we can do can help or hurt this fact. What is the Biblical basis for this?
Unconditional Election refers to
God's choosing of people to be the object of his grace or otherwise fulfill his purpose.
There are a couple of versus specifically that are associated with this.
For those whom he foreknew he also predestined to be conformed to the image of his Son, in order that he might be the firstborn among many brothers.
and 1 Peter 1:1-2
1 Peter, an apostle of Jesus Christ,
To those who are elect exiles of the dispersion in Pontus, Galatia, Cappadocia, Asia, and Bithynia, 2 according to the foreknowledge of God the Father, in the sanctification of the Spirit, for obedience to Jesus Christ and for sprinkling with his blood:
May grace and peace be multiplied to you.
The idea is that God knew these people beforehand and chose these people to be saved. It's a similar concept to predestination.
The Calvinist argument can be found here
Richard's answer doesn't address the primary issue of Unconditional Election, which is that God's election of people to salvation is based entirely on himself, not on any criteria about the person.
e.g. Many people ask me if God didn't simply elect the people that he knew would accept salvation.
The Doctrine of Total Depravity implies that there is no merit to base any criteria or condition on.
More directly, in Romans 3:27-30 Paul makes it very clear that there can be no boasting in our salvation. If God had any preference for one type of person over another, then we could use our salvation to boast as evidence of that. But since there can be no boasting, then God must not prefer one type of person over another.
Similarly, in Romans 9:10-18, Paul argues that God makes his decision with no dependence on the man or woman, but entirely on himself. He references God's choosing Jacob over Esau in verse 11, "though they were not yet born and had done nothing either good or bad—in order that God's purpose of election might continue". This makes it clear that God's reasons for choosing Jacob were for "God's purpose of election", not in Jacob's worth or actions.
Unconditional election is elegantly and concisely defined in the Belgic Confession, article 16:
We believe that—all Adam’s descendants having thus fallen into perdition and ruin by the sin of Adam—God showed himself to be as he is: merciful and just.
God is merciful in withdrawing and saving from this perdition those who, in the eternal and unchangeable divine counsel, have been elected and chosen in Jesus Christ our Lord by his pure goodness, without any consideration of their works.
God is just in leaving the others in their ruin and fall into which they plunged themselves.
You can read fuller definitions and defenses of the doctrine in the first section of the Canons of Dort and in chapter 3 of the Westminster Confession of Faith, but the essentials can be identified pretty clearly:
God could justly condemn all men for their sins, but he chose to elect some to salvation and elect others to their due condemnation.
God chose the elect for salvation not based on anything good he foresaw in them, but according to the good pleasure of his will.
God's decree of election is eternal and unchangeable.
Romans 8, 9, and 11 spell out the doctrine possibly more clearly than anywhere else. Romans 8:28-30 lays out what is often called the "golden chain of salvation":
And we know that all things work together for good for those who love God, who are called according to his purpose, because those whom he foreknew he also predestined to be conformed to the image of his Son, that his Son would be the firstborn among many brothers and sisters. And those he predestined, he also called; and those he called, he also justified; and those he justified, he also glorified.
Arminians believe that "foreknew" means that God first foresaw who would love God, and then predestined them to salvation. But Calvinists believe that this is absurd: to love God, one first has to "be conformed to the image of his Son." So they believe that foreknowledge refers to God "knowing" or loving his people from eternity. This sense of "know" can be seen in Jeremiah 1:5:
Before I formed you in your mother's womb I chose [Hebrew: "knew"] you. Before you were born I set you apart. I appointed you to be a prophet to the nations.
The inseparability of the links in the "golden chain" can be demonstrated by 1 John 2:19:
They went out from us, but they did not really belong to us, because if they had belonged to us, they would have remained with us. But they went out from us to demonstrate that all of them do not belong to us.
The eternality of the election decree can also be seen in these verses:
For he chose us in Christ before the foundation of the world that we may be holy and unblemished in his sight in love. He did this by predestining us to adoption as his sons through Jesus Christ, according to the pleasure of his will— to the praise of the glory of his grace that he has freely bestowed on us in his dearly loved Son.
2 Thessalonians 2:13-14
But we ought to thank God always for you, brothers and sisters loved by the Lord, because God chose you from the beginning for salvation through sanctification by the Spirit and faith in the truth. 14 He called you to this salvation through our gospel, so that you may possess the glory of our Lord Jesus Christ.
When the Gentiles heard this, they began to rejoice and praise the word of the Lord, and all who had been appointed for eternal life believed.
Romans 9 has a lot to say on the subject of election:
6 It is not as though the word of God had failed. For not all those who are descended from Israel are truly Israel, 7 nor are all the children Abraham’s true descendants; rather “through Isaac will your descendants be counted.” 8 This means it is not the children of the flesh who are the children of God; rather, the children of promise are counted as descendants. 9 For this is what the promise declared: “About a year from now I will return and Sarah will have a son.” 10 Not only that, but when Rebekah had conceived children by one man, our ancestor Isaac— 11 even before they were born or had done anything good or bad (so that God’s purpose in election would stand, not by works but by his calling)— 12 it was said to her, “The older will serve the younger,” 13 just as it is written: “Jacob I loved, but Esau I hated.”
So not all members of national Israel are part of the true Israel, and those of the true Israel are chosen before they are even born. Paul continues:
14 What shall we say then? Is there injustice with God? Absolutely not! 15 For he says to Moses: “I will have mercy on whom I have mercy, and I will have compassion on whom I have compassion.” 16 So then, it does not depend on human desire or exertion, but on God who shows mercy. 17 For the scripture says to Pharaoh: “For this very purpose I have raised you up, that I may demonstrate my power in you, and that my name may be proclaimed in all the earth.” 18 So then, God has mercy on whom he chooses to have mercy, and he hardens whom he chooses to harden.
So according to God's good pleasure he has mercy on some and leaves others in their wickedness. This is confirmed in other verses:
The Lord works everything for its own ends—even the wicked for the day of disaster.
1 Peter 2:8b
They stumble because they disobey the word, as they were destined to do.
This comports well with how Paul continues his argument in Romans 9:
21 Has the potter no right to make from the same lump of clay one vessel for special use and another for ordinary use? 22 But what if God, willing to demonstrate his wrath and to make known his power, has endured with much patience the objects of wrath prepared for destruction?
In Romans 11, Paul begins by speaking of the "remnant" that God preserved during the time of Elijah:
5 So in the same way at the present time there is a remnant chosen by grace. 6 And if it is by grace, it is no longer by works, otherwise grace would no longer be grace. 7 What then? Israel failed to obtain what it was diligently seeking, but the elect obtained it. The rest were hardened, 8 as it is written, “God gave them a spirit of stupor, eyes that would not see and ears that would not hear, to this very day.” [Quoting Deuteronomy 29:4]
9 And David says, “Let their table become a snare and trap, a stumbling block and a retribution for them; 10 let their eyes be darkened so that they may not see, and make their backs bend continually.” [Quoting Psalm 69:22-23]
As we've already seen, the idea of God's salvation not being based on any good that he foresees goes back to the Old Testament. God says of Israel:
It is not because you were more numerous than all the other peoples that the Lord favored and chose you—for in fact you were the least numerous of all peoples. Rather it is because of his love for you and his faithfulness to the promise he solemnly vowed to your ancestors that the Lord brought you out with great power, redeeming you from the place of slavery, from the power of Pharaoh king of Egypt.
I made myself available to those who did not ask for me; I appeared to those who did not look for me. I said, ‘Here I am! Here I am!’ to a nation that did not invoke my name.
It's all over the New Testament as well. Faith is seen as a gift:
You did not choose me, but I chose you and appointed you to go and bear fruit, fruit that remains, so that whatever you ask the Father in my name he will give you.
44 No one can come to me unless the Father who sent me draws him, and I will raise him up at the last day. ... 64 But there are some of you who do not believe. ... 65 Because of this I told you that no one can come to me unless the Father has allowed him to come.
2 Timothy 1:9
He is the one who saved us and called us with a holy calling, not based on our works but on his own purpose and grace, granted to us in Christ Jesus before time began.
2 Peter 1:1
From Simeon Peter, a slave and apostle of Jesus Christ, to those who through the righteousness of our God and Savior, Jesus Christ, have been granted a faith just as precious as ours.
For it has been granted to you not only to believe in Christ but also to suffer for him.
We've already seen verses that speak of "hardening" the reprobate. God actively hides knowledge from some, just as he actively reveals it to others, all for his glory:
At that time Jesus said, “I praise you, Father, Lord of heaven and earth, because you have hidden these things from the wise and intelligent, and revealed them to little children. Yes, Father, for this was your gracious will.
You have been given the opportunity to know the secrets of the kingdom of heaven, but they have not.
1 Corinthians 1:26-31
Think about the circumstances of your call, brothers and sisters. Not many were wise by human standards, not many were powerful, not many were born to a privileged position. But God chose what the world thinks foolish to shame the wise, and God chose what the world thinks weak to shame the strong. God chose what is low and despised in the world, what is regarded as nothing, to set aside what is regarded as something, so that no one can boast in his presence. He is the reason you have a relationship with Christ Jesus, who became for us wisdom from God, and righteousness and sanctification and redemption, so that, as it is written, “Let the one who boasts, boast in the Lord.”