This is often called the "Reformed" view, or "Calvinist" perspective. Basically, people who hold this view believe that God has predetermined those people who will be saved. In other words, he has elected some to spend eternity with him in heaven. This election has no basis on the individual's behavior or morality, but is wholly based on the grace of God. Because God is completely holy, and because we are utterly sinful (totally depraved), we cannot choose to follow God or believe in him of our own will. That is to say, we are too sinful to humble ourselves, repent of our sins, and place our faith in Christ. This faith must be a gift from God, flowing out of his grace. The elect are those to whom this faith has been given.
This is often called the "Arminian" perspective. People who believe in free will understand humans to have a choice in whether they repent of their sins and place their faith in Jesus Christ. God's will is for all people to be saved, but, out of his gracious humility, he has left the choice of faith to us. He calls us to follow him, and we can freely embrace him or reject him. God compels no one to choose him against their will; rather, he honors human beings as free moral agents created in his image. God's grace comes as both a gift and an offer, the intended response to which is faith. Everyone who accepts this offer will be saved.
These are two very brief sketches, and I hope that I've done justice to each perspective. I fall into the free will camp, but have often felt the weakness of the Arminian position in its poor explanation (or total lack thereof) of election. In what follows I'd like to begin an attempt at explaining what election means.
In order to understand biblical election, we have to set aside the medieval notion of the subject, namely that God has elected certain individuals for salvation and others for damnation. This is unhelpful and anachronistic. Understanding election means going back to the source, and discovering in the Scriptures what is meant by this controversial term.
What did election mean for Jesus? In fact, election was a core tenant of Jesus' faith. As N.T. Wright has shown, Jesus fully believed in election, though not in the same way in which we define it today. Jesus believed that the creator God has intended, from the very beginning, to address and deal with the problems of creation through Israel. When everything went wrong in Adam--when sin and death entered the world through him--God intended to set everything aright through Abraham and his descendants.
Israel was chosen by God to be the instrument by which sin and death would be undone, and everything in creation would be set to rights. Israel was The Elect. Unfortunately, Israel failed to live up to their high calling. They failed to be The Elect. In steps Jesus, to be the Israel that Israel could never be, and to do what Israel was always meant to do--set the world to rights by atoning for sin and conquering death. In other words, Jesus is The Elect. He has done what The Elect were elected to do.
Any understanding of election must begin with Israel and move then to Jesus. This, rather than individualistic predestination, is the biblical view of election.
Because Jesus is The Elect, all who have faith in him are The Elect in him. Election is not an arbitrary divine choice, but rather a new reality and identity bestowed on all who obey Jesus' command to believe in him. Salvation is the gift of God, and election is the new reality brought about by the reception of that gift.
God has not, as some would contend, elected some for salvation and, therefore, others for damnation. Instead, God has elected his Son to right the wrongs of the world by atoning for sin and conquering death through his own crucifixion and resurrection. All who confess Jesus as King are saved and become a part of The Elect in Christ because they have become a part of Christ through faith. Their task becomes the task of The Elect: Announcing to the world that Jesus Christ has atoned for sin through his death and conquered death through his resurrection.
Biblical election, therefore, is a new reality that comes with a very old mission. Live, then, as The Elect, announcing and enacting the gospel of Jesus Christ.