A related, updated, and more detailed response to this question is found here
Question: Do Calvinists believe that Jesus didn't die for all, just the elect, and if so is there any scripture to back it up.
Answer: Calvinism is perhaps the most misunderstood and misrepresented theological belief system found within modern Christianity. Some of this stems from simplistic representations, and caricatures, held up by those who would casually, or callously, dismiss the real tenants of Calvinism without detailed and biblical examination.
The complete sovereignty of God, over the affairs of man, is at the heart of what has become known as Calvinism. Virtually all of the early Protestant reformers, including Martin Luther, held to the sovereignty of God in regards to salvation (read Luther's "Bondage of the Will" for a better understanding of his views). At issue was the question of grace versus works. Is some work or effort of man necessary in salvation, or is salvation completely and solely from God and by God?
This completely unoriginal paragraph summaries Calvinism...
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.
Some, today, have distanced themselves from the term Calvinist because of the distortions that now abound (including the historic, yet incorrect, hyper-calvinism). Many prefer to merely say that they hold to the doctrine of grace. Even the usual TULIP acrostic, directly associated with Calvinism (which we'll get to in a moment), has other variations including GRACE...
G - God's Sovereign Grace
Many denominations, which once held to Calvinism, have dropped or moderated their views over the years; mostly due to general ignorance, or a general unwillingness to study these matters from scriptures. For example, early Baptist confessions of faith (including the London Confession of Faith and the New Hampshire Confession of Faith) reflect their Calvinist convictions. Charles Spurgeon professed himself a Calvinist and reflected such in both his writings and Catechism. Even churches that followed the classic Westminster Confession of Faith, including some Presbyterian churches, have since changed their doctrine.
Regardless of what any church, or denomination, or individual teaches, we are to be followers of Jesus Christ and students of the Word. We are not followers of John Calvin or any other Christian -- if they held (or hold) to a truth or doctrine that can be found in scriptures, so should we. Should any doctrine be found to be otherwise, we should be willing to discard that teaching as error. This being said, each individual should make a thorough and unbiased search of scriptures to establish a correct understanding of the sovereignty of God and a true understanding of grace. The Holy Spirit will guide you in all truth.
For the record, in a quick summary, here are the five primary points of Calvinism given in their classic TULIP acrostic...
T - Total Depravity
The following descriptions of each point are from the appendix of the book "The Reformed Doctrine of Predestination" by Loraine Boettner -- excluding the last point "P", which I revised, as I felt it was inadequate in summarizing the full reformed view of perseverance.
Because of the fall in Adam, man is unable of himself to savingly believe the Gospel. The sinner is spiritually dead, blind, and deaf to the things of God; his heart is deceitful and desperately corrupt. His will is not free but is in bondage to his sinful nature. He is as spiritually dead and estranged from God as are the fallen angels. Therefore he will not, indeed cannot, choose good over evil in the spiritual realm. Consequently it takes regeneration by which the Spirit brings a sinner to Christ - it takes regeneration by which the Spirit brings a sinner from spiritual death to spiritual life and gives him a new nature. Faith is not something man contributes to salvation but is itself a part of God's gift of Salvation - it is God's gift to the sinner , not the sinner's gift to God.
John 6:44; Romans 5:6 and 8:7,8; Ephesians 2:1,5; Colossians 2:13; Titus 3:3-5
God's choice of certain individuals unto salvation, before the foundation of the world, rested solely on His own Sovereign Will. His choice of particular sinners was not based on any foreseen response or obedience on their part, such as faith, repentance, etc. On the contrary, God gives faith and repentance to each individual whom He selected. These acts are the result, not the cause, of God's choice. Election therefore was not determined by or conditioned upon an virtuous quality or act foreseen in man. Those whom God sovereignty elected He brings through the power of the Spirit to a willing acceptance of Christ. Thus God's choice of Christ, is the ultimate cause of salvation.
Romans 9:11; 11:5,7; 11:28; 8:38-39; 1 Thessalonians 1:4; 2 Thessalonians 2:13; 14:22; Peter 1:10; Ephesians 1:3-14; 2 Timothy 1:9,10
Limited Atonement (also known as Particular Atonement)
Christ's redeeming work was intended to save the elect only and actually secured salvation for them. His death was a substitutionary endurance of the penalty of sin in the place of certain specified sinners. In addition to putting away the sins of His people, Christ's redemption secured everything necessary for their salvation, including faith which unites them to Him. The gift of faith is infallibly applied by the Spirit to all for whom Christ died, thereby guaranteeing their salvation. [Note: For those who get hung-up on the word, "Limited", this has also been described as; "Christ's death being sufficient for all, yet efficient for some."]
Matt. 1:2; 1 John 6:37-40, 10:14-16, 26-28, 15:13-14; Acts 20:28; Heb. 10:14; Rom. 8:31-34; 1 Cor. 8:11; Eph. 5:25-27; 1 John 4:10-11
In addition to the outward general call to salvation which is made to everyone who hears the Gospel, the Holy Spirit extends to the elect a special inward call that inevitably brings them to salvation. The external call (which is made to all without distinction) can be, and often is, rejected, whereas the internal call (which is made only to the elect) cannot be rejected; it always results in conversion. By means of this special call the Spirit irresistible draws sinners to Christ. He is not limited in His work of applying salvation by man's will, nor is He dependent upon man's cooperation for success. The Spirit graciously causes the elect sinner to cooperate, to believe, to repent, to come freely and willingly to Christ. God's grace, therefore, is invincible; it never fails to result in the salvation of those to whom it is extended.
Not the labors of my hands
Eph. 1:19, 20; Ezek. 11:19; I Cor. 4:7; John 17:2; John 5:21; Acts 13:48
Perseverance Of The Saints
All who were chosen by God, redeemed by Christ, and given faith by the Spirit are eternally saved. By the power of the indwelling Spirit their lives are changed; enabling them to bear good fruit. They are kept in faith by the power of Almighty God and thus persevere to the end.
John 6:37-39; Phil. 1:6; 1 Thess. 5:23-24; 2 Tim. 4:18; 1 Peter 1:23
A final note. A common form of hyper-calvinism teaches that since salvation is completely God's choice, and by His doing, therefore the church (and all Christians) have no responsibility for missions, evangelism, etc., as "God will save them if He wants to." This is a heretical concept that cannot be reconciled with scripture and true Calvinism. The true Calvinist believes that though God chooses and enables, all Christians are commanded to preach and witness, and call everyone to repentance and salvation. Another Calvinism distortion is that people can live lives of sin (never repenting or bearing fruit) and God will save them if He desires. This becomes an excuse for lawlessness and again twists the true belief of Calvinism -- that saving faith effects change in all believers.
Links to check out...
Calvinistic Confessions of Faith...