Basically, Calvinism is known by an acronym: T.U.L.I.P.
Total Depravity (also known as Total Inability and Original Sin)
Unconditional Election
Limited Atonement (also known as Particular Atonement)
Irresistible Grace
Perseverance of the Saints (also known as Once Saved Always Saved)
These five categories do not comprise Calvinism in totality. They simply represent some of its main points.
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: Mans 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 mans 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 s fallen nature we are born again not by our own will but Gods 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).
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. (Rom. 9:15, 21).
(On a personal note... I don't really believe personally that God "elects" or chooses anyone in the hell, But people choose that themselves, by rejecting the Gospel of Jesus Christ. Not to contradict myself... But I do believe we have a choice in that matter.)
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).
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 mans will, but by Gods.
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.