The important first step with any theological position or term is to understand what question or set of questions it is a part of. Predestination has to do with questions regarding salvation. In general, it is part of answering the question: "How are we saved?"
How are we saved? This is not a simple question to answer. However, in it's most general perspective we can say we are saved by the grace of God. This grace chose us (or predestined us) from the beginning to live and reign with him in all eternity. But predestination involves more than simply God's choosing us. It also involves the movements of God's grace, which move us to seek God and to find him in faith, in baptism and in the sacramental life of the church. This predestination also involves grace to help us to persevere until the end, providing us with the opportunities and the means to repent and return should we sin and fall away from God. So, in a nutshell, predestination is God's choosing us in love, and giving us the means to attain salvation through a life graced by faith, sacraments, the church, repentance, and perseverance.
It is important to note that salvation occurs first and foremost as God's initiative. The term predestination emphasizes this initiative at every step of our journey. Without God's initiative, we would fail. If God did not choose us, we could not be saved. If God does not draw us through grace and lead us to faith, we could not be saved.If God does not give us faith, we could never attain it on our own, and we could not be saved. If God does not cleanse us of our sins we could not be saved. If God does not bring us into the life of the church, we could not be saved. If God did not draw us back after sin, give us the grace of repentance, and give us the grace of perseverance, we could not be saved. It is even impossible, ultimately, to lead a moral life without God's assistance. Everything is grace, and without it we would fail.
However, problems arise: does God predestine all people to be saved? Catholics, I think would argue yes. In his encyclical Redemptoris Missio, Pope John Paul II writes: "Since salvation is offered to all, it must be made concretely available to all. . . . This grace comes from Christ; it is the result of his Sacrifice and is communicated by the Holy Spirit. It enables each person to attain salvation through his or her free cooperation." [Section 10.1] Further, Catholics teach that "God predestines no one to go to hell" [See the Catechism of the Catholic Church, section 1037]. Hence, Catholics argue that salvation is made available to all and willed for all, even those who are not explicitly part of the Church; all are thus predestined for salvation.
Another problem then arises: how does it happen that some will not be saved? This is the problem with which Calvinist theology wrestles. If God, in fact, predestines all for salvation, how is it possible for anyone not to be saved? Catholics would answer this question by saying that human beings are given the freedom to choose or refuse God's offer of salvation; or to use the Pope's term above, they are free to cooperate with God's grace or not. In so doing, Catholics do not minimize the need for God's grace, which makes it possible for human beings to cooperate in the first place. However, they recognize that God's plan includes the possibility that human beings could throw away the gift they have been given.
This answer dissatisfies Calvinists. This is the reason for their dissatisfaction: It would seem, that if human beings are able to frustrate God's will and the power of God's grace in bringing all people to salvation, then God's grace and will are not all powerful. Human decisions have the power to overturn God's plans and even God's actions. Thus, human beings, would seem to be more powerful than God in this respect. They have the power to frustrate God's plan and power. This possibility is untenable for many Calvinists. Calvinism grew out of that branch of Catholicism (Franciscan theology) which emphasized how God was all powerful, and that whatever God willed came to pass. It was impossible for any creature to resist God's will or to frustrate God's plan. Thus, for a Calvinist, if any people are not saved, it is not merely because they chose not to be saved. It is also because God willed or predestined it, that they would not be saved.
Clearly, this conclusion does not sit well with Catholics. But the objections of the Calvinists are important theological points, and not to be easily dismissed. We often call God "All Powerful" and "Almighty" ourselves. How is it possible that human beings can frustrate the designs of an all powerful God? Some may argue perhaps that ultimately those who are not saved do not frustrate God's designs because they demonstrate God's justice. This is the Calvinist answer, and it shows that those who reject God have a place in God's plan. The problem with this answer, however, is that it does not clearly establish that God does not want the sinner to die in sin; God does not will the damnation of anyone. If the sinner does not repent and return to God, then God's plan is frustrated with respect to that person. It would seem, then, that God is not all powerful.
Catholics, I think, would then argue: yes, God has freely chosen to limit the sphere of his own power and freedom. Being all powerful, he and he alone has the power to limit the sphere in which his will is effective. By his own will, God freely chooses to allow individual human beings the freedom to choose or refuse him. Thus, Catholics would argue, that in creating free will, God has chosen to limit the realm of his power, creating a being that can frustrate God's will, with respect to that particular person. However, this person cannot frustrate God's overall plan, and this is the truth Calvinists want to protect.
So this is the problem of predestination in summary. There are, as you can imagine, several other wrinkles to the mix. The interplay between grace, free will, and the role of the church and sacraments raises many, many more questions. However, as I understand it, what I have written are some of the principal parts of the difficulties between Calvinists and Catholics on this point of predestination.