According to Calvinism:
Salvation is accomplished by the almighty power of the Triune God. The Father chose a people, the Son died for them, the Holy Spirit makes Christ's death effective by bringing the elect to faith and repentance, thereby causing them to willingly obey the gospel. The entire process (election, redemption, regeneration) is the work of God and is by grace alone. Thus God, not man, determines who will be the recipients of the gift of salvation.
According to Arminianism:
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.
The five points of Calvinism and the Arminian perspective:
1) Total Depravity:
Sin has affected all parts of man. The heart, emotions, will, mind, and body are all affected by sin. We are completely sinful. We are not as sinful as we could be, but we are completely affected by sin.
The doctrine of Total Depravity is derived from scriptures that reveal human character: Man’s heart is evil (Mark 7:21-23) and sick (Jer. 17:9). Man is a slave of sin (Rom. 6:20). He does not seek for God (Rom. 3:10-12). He cannot understand spiritual things (1 Cor. 2:14). He is at enmity with God (Eph. 2:15). And, is by nature a child of wrath (Eph. 2:3). The Calvinist asks the question, "In light of the scriptures that declare man’s true nature as being utterly lost and incapable, how is it possible for anyone to choose or desire God?" The answer is, "He cannot. Therefore God must predestine."
Calvinism also maintains that because of our fallen nature we are born again not by our own will but God’s will (John 1:12-13); God grants that we believe (Phil. 1:29); faith is the work of God (John 6:28-29); God appoints people to believe (Acts 13:48); and God predestines (Eph. 1:1-11; Rom. 8:29; 9:9-23).
2) Unconditional Election:
God does not base His election on anything He sees in the individual. He chooses the elect according to the kind intention of His will (Eph. 1:4-8; Rom. 9:11) without any consideration of merit within the individual. Nor does God look into the future to see who would pick Him. Also, as some are elected into salvation, others are not (Rom. 9:15, 21).
3) Limited Atonement:
Jesus died only for the elect. Though Jesus’ sacrifice was sufficient for all, it was not efficacious for all. Jesus only bore the sins of the elect. Support for this position is drawn from such scriptures as Matt. 26:28 where Jesus died for ‘many'; John 10:11, 15 which say that Jesus died for the sheep (not the goats, per Matt. 25:32-33); John 17:9 where Jesus in prayer interceded for the ones given Him, not those of the entire world; Acts 20:28 and Eph. 5:25-27 which state that the Church was purchased by Christ, not all people; and Isaiah 53:12 which is a prophecy of Jesus’ crucifixion where he would bore the sins of many (not all).
4) Irresistible Grace:
When God calls his elect into salvation, they cannot resist. God offers to all people the gospel message. This is called the external call. But to the elect, God extends an internal call and it cannot be resisted. This call is by the Holy Spirit who works in the hearts and minds of the elect to bring them to repentance and regeneration whereby they willingly and freely come to God. Some of the verses used in support of this teaching are Romans 9:16 where it says that "it is not of him who wills nor of him who runs, but of God who has mercy"; Philippians 2:12-13 where God is said to be the one working salvation in the individual; John 6:28-29 where faith is declared to be the work of God; Acts 13:48 where God appoints people to believe; and John 1:12-13 where being born again is not by man’s will, but by God’s.
5) Perseverance of the Saints:
You cannot lose your salvation. Because the Father has elected, the Son has redeemed, and the Holy Spirit has applied salvation, those thus saved are eternally secure. They are eternally secure in Christ. Some of the verses for this position are John 10:27-28 where Jesus said His sheep will never perish; John 6:47 where salvation is described as everlasting life; Romans 8:1 where it is said we have passed out of judgment; 1 Corinthians 10:13 where God promises to never let us be tempted beyond what we can handle; and Phil. 1:6 where God is the one being faithful to perfect us until the day of Jesus’ return.