Did you know that Catholics believe in predestination? Yep, we do. But not the Puritanical version of predestination. So let's start from there.
Predestination is the belief that God foreordains things. In the Puritanical definition it includes foreordaining the elect (a certain group of peoples) to heaven or salvation. Puritans held that nobody knew if they were the elect, but with the elect they behaved well. So everybody behaved themselves. They also said that it didn't matter what you did right or wrong if you were chosen by God, then you were saved. This is like my piece of Eternal Security, but not quite.
As I said, Predestination also involves God foreordaining other things.
Here is what the CCC has to say about predestination (incidentally you can find the CCC online so if you want to check that I'm accurate you can see for yourself)
CCC 600: To God, all moments of time are present in their immediacy. When therefore he establishes his eternal plan of "predestination," he includes it in each person's free response to his grace: "In this city, in fact, both Herod and Pontius Pilate, with the Gentiles and the people of Israel, gathered together against your holy servant Jesus, whom you anointed, to do whatever your hand and your plan had predestined to take place." For the sake of accomplish his plan of salvation, God permitted the acts that flowed from their blindness.
CCC 1037: God predestines no one to go to hell; for this a willful turning away from God (a mortal sin) is necessary, and persistence in it until the end.
So Catholics believe that an over-all plan has been predestined. We all are able to receive grace freely and God calls us to do so. However, we are allowed to turn away willingly, but this does not mean that God predestines us to hell. His plan is to save everyone. What the details of his plans for all of us are, no one knows. It's a great mystery, but suffices to say that he loves us and intends to save us all. The elect, I believe, are those who do not turn away from God. I don't believe that the elect are a foreordained few.
Biblical references to Christ saving us all: Romans 2:17-19, 1 Cor 15:21-22, 1 Timothy 2:6, 1 Timothy 4:10
Links to sites explaining Catholic predestination: The Misunderstanding of Catholic Predestination