The debate over how to understand Romans 9 largely stems from the debate over free will and predestination. So we are all on the same page here is a helpful summary of Arminianism that I found online. Calvinism will come tomorrow.Free-Will or Human Ability
Although human nature was seriously affected by the fall, man has not been left in a state of total spiritual helplessness. God graciously enables every sinner to repent and believe, but He does not interfere with man's freedom. Each sinner possesses a free will, and his eternal destiny depends on how he uses it. Man's freedom consists of his ability to choose good over evil in spiritual matters; his will is not enslaved to his sinful nature. The sinner has the power to either cooperate with God's Spirit and be regenerated or resist God's grace and perish. The lost sinner needs the Spirit's assistance, but he does not have to be regenerated by the Spirit before he can believe, for faith is man's act and precedes the new birth. Faith is the sinner's gift to God; it is man's contribution to salvation.
Conditional Election
God's choice of certain individuals unto salvation before the foundation of the world was based upon His foreseeing that they would respond to His call. He selected only those whom He knew would of themselves freely believe the gospel. Election therefore was determined by or conditioned upon what man would do. The faith which God foresaw and upon which He based His choice was not given to the sinner by God (it was not created by the regenerating power of the Holy Spirit) but resulted solely from man's will. It was left entirely up to man as to who would believe and therefore as to who would be elected unto salvation. God chose those whom He knew would, of their own free will, choose Christ. Thus the sinner's choice of Christ, not God's choice of the sinner, is the ultimate cause of salvation.
Universal Redemption or General Atonement
Christ's redeeming work made it possible for everyone to be saved but did not actually secure the salvation of anyone. Although Christ died for all men and for every man, only those who believe on Him are saved. His death enabled God to pardon sinners on the condition that they believe, but it did not actually put away anyone's sins. Christ's redemption becomes effective only if man chooses to accept it.
The Holy Spirit Can Be Effectually Resisted
The Spirit calls inwardly all those who are called outwardly by the gospel invitation; He does all that He can to bring every sinner to salvation. But inasmuch as man is free, he can successfully resist the Spirit's call. The Spirit cannot regenerate the sinner until he believes; faith (which is man's contribution) proceeds and makes possible the new birth. Thus, man's free will limits the Spirit in the application of Christ's saving work. The Holy Spirit can only draw to Christ those who allow Him to have His way with them. Until the sinner responds, the Spirit cannot give life. God's grace, therefore, is not invincible; it can be, and often is, resisted and thwarted by man.
Falling from Grace
Those who believe and are truly saved can lose their salvation by failing to keep up their faith, etc. All Arminians have not been agreed on this point; some have held that believers are eternally secure in Christ - that once a sinner is regenerated, he can never be lost.
In sum, "Free Willers" or Arminians believe that salvation is accomplished through the combined efforts of God (who takes the initiative) and man (who must respond) - man's response being the determining factor. God has provided salvation for everyone, but His provision becomes effective only for those who, of their own free will, "choose" to cooperate with Him and accept His offer of grace. At the crucial point, man's will plays a decisive role; thus man, not God, determines who will be recipients of the gift of salvation.
