We could summarize the Church’s teaching as follows: God has perfect foreknowledge of who will be elect (which is the same thing as to say that he predestines them). However, although salvation is entirely His work, He works in a manner that does not interfere with man’s freedom (including man’s freedom to reject God) in any way.
The Parameters that All Catholics Must Agree On
To build on Geremia’s answer, which appropriately references the Council of Trent, it should be noted that it is not predestination as such (predestination is a concept found in the epistles of Paul—e.g., Romans 8:28-30), but how that concept is interpreted, that can be a problem.
This is a very complex issue, which has had a number of interpretations even among Catholics. In the first part of this answer, I will lay out the basic parameters that the Church sets.
I should point out right from the outset that predestination is defined (at least by the dominant theological school, Thomism) as divine foreknowledge of the salvation of the elect.
Unde manifestum est quod praedestinatio est quaedam ratio ordinis aliquorum in salutem aeternam, in mente divina existens.
Therefore, it is evident that predestination is a certain notion of the ordering of some persons to eternal life, existing in the divine mind (Summa theologica [S.Th.] I, q. 23, a. 2, responsum; my translation; the public-domain translation from the English Dominican province, found at the source I linked to, is somewhat misleading at this point).
In other words, God already knows from all eternity who will be saved (i.e., who is predestined) and who will not be saved (i.e., who is reprobate).
In essence, the parameters are as follows:
Salvation is the work of God, and it is mediated by grace. Only those (and all those) who persevere in possessing sanctifying grace are saved, because sanctifying grace is actually the foretaste or beginning of the perfect union with God that the saints enjoy in Heaven. (See Catechism of the Catholic Church [CCC], 1996.)
Even the beginning of man’s life of grace is the work of God. Man can do nothing—not even begin the journey toward God—without the help of God. (See CCC 2001. Generally, we call the concrete actions that God employs to bring a person into friendship with Him actual graces, to distinguish them from sanctifying grace, which is habitual and stable, as explained in CCC 2000.)
Nevertheless, the action of God on a person can only be done with the person’s consent, unless the person is insufficiently mature or incapacitated, like young children and the mentally handicapped. (See CCC 2002.)
When a person receives sanctifying grace, by that very fact he is also justified—that is, rendered truly righteous and pleasing before God—and sanctified—that is, made holy. (See CCC 1996 and 1999.) Grace is also healing: it repairs the damage caused by his previous sins, especially the resulting separation from God. (See CCC 1990.)
(Sanctifying grace, incidentally, is generally received for the first time at Baptism, provided the recipient does not place any obstacles. How sanctifying grace works in the unbaptized is rather a mystery, but it must be received at some point before death, if the person is to be saved.)
Once a person has received sanctifying grace, he can lose it by committing a grave (mortal) sin, provided, of course, that he has sufficient knowledge, is free to perform the action, and gives the full consent of his will. (See CCC 1961 and 1857.) He can recover that grace by repenting, motivated out of supernatural love. (See CCC 1451-1454.) The ordinary path to that repentance, for the baptized (obligatory for all Catholics, barring some kind of impossibility), is the Sacrament of Reconciliation. (See CCC 1456.)
God, being omniscient, has full foreknowledge of who will be saved. However, that knowledge in no way impedes a person’s freedom to opt for that salvation or not. (See CCC 600.)
How to Reconcile Predestination and Freedom
Within the Catholic Church, a number of theological schools have arisen regarding how God’s foreknowledge can be reconciled with man’s freedom of choice. As long as they stay within the Church’s parameters, the faithful are free to adhere to any school they find convincing.
I will offer here some reflections from St. Thomas Aquinas that help to see how this works. (It is important not to confuse Aquinas’ own position with that of so-called “Thomistic school,” which in reality was founded by a Dominican named Domingo Bañez, some 350 years after St. Thomas.)
A word on Contingency, Necessity, and God’s Creative Action
God is omniscient and omnipotent, and so it follows that everything that God wills is accomplished, in exactly the way that He wants it.
Although God cannot fail in His action, nevertheless, God can will for certain things to come to pass in a necessary way, and others in a contingent way.
In Aquinas’ lingo, something is “necessary” when it cannot possibly be otherwise. For example, if a kettle of water is placed on the fire, it is necessary (assuming normal atmospheric pressure and so forth) that the water should eventually boil. The water has no say in the matter. (In Aquinas, there are various kinds of necessity: the kind that interests us at the moment is physical necessity, or the necessity of coercion.)
On the other hand, suppose that the government orders everyone to file his tax return on April 15. Whether everyone actually does so is up to the individual. Sure, there are bad consequences for not complying, but at the end of the day the IRS cannot force everyone to turn in his tax return on time. In other words, actually turning in your tax return is contingent: it is entirely possible (indeed likely) that someone will not comply.
In particular, man, insofar as his will — or intellectual appetite — is concerned, is not bound by a physical necessity. Given a limited, created good, he may opt for it, or not. (Only the Beatific Vision is capable of obliging the will.) This contingency is, of course, the foundation of man’s freedom of choice.
(See the responsa of S.Th. q. 19, a. 3, and q. 82, articles 1-2, for a fuller treatment of necessity and contingency and how they apply to the human will.)
Hence, although God knows what each man will do, even in the future, He has created man in such a way that man is free to choose the course of action that he wishes: even sin, and even eternal condemnation, if he so desires.
It should be noted, therefore, that sin and condemnation are not “failures” on God’s part, but only on man’s part. God is fully aware that his creatures are capable of opting away from Him: that ability is part of the marvelous perfection with which God endowed us.
If God Knows Whether We Will Be Saved, How Do We Remain Free?
It is important to recall that man lives immersed in time, but not God. Man always lives in the present moment. From man’s perspective, the future does not exist yet, and the past exists no longer. However, God—the creator of time—exists entirely outside of time. He sees all of history simultaneously in an eternal present.
We human beings have no trouble knowing about contingent actions that are in the present (at least, those that are readily visible, not the interior dispositions of a person’s soul, obviously). For example, the fact that I am seated right now at the computer is contingent: it depends on me; I could just as easily be standing up right now, or taking a walk. If the reader were to see me right now, he would see that I am seated.
On the other hand, for us, future contingent actions are a mystery. The reader can only conjecture as to what I will be doing an hour from now. (And not even I can be absolutely sure.)
Crucially, however, the fact that the reader knows that I am seated right now does not compel me to sit. The fact that I am seated remains contingent; it does not suddenly become necessary when someone discovers it.
God does not have to make any conjectures about what will happen in the future, since he sees everything in an eternal present, as I mentioned. What He created contingent, moreover, he does not transform into something necessary by the mere fact that He knows it. (After all, contingent is exactly how He wanted it to be.) Hence, in particular, the fact that God knows we will be saved (or not) does not in any way prevent that decision from being a contingent one (one that is up to us to decide).
(This very argument is made in the responsum to S.Th. I. q. 14, a. 13, which regards God’s knowledge of future contingents.)
In summary: God created man with the ability to perform contingent actions (actions that are up to us). God, being omnipotent, does indeed have foreknowledge of who will be saved—that is, He predestines some to Heaven, and is even the cause of their salvation—but this foreknowledge (which, for God, takes place in an eternal present) does not take away at all from the contingency of our actions. His knowledge does not make them necessary; he does not compel us to be saved (or not).