I wrote up a brief statement outlining some of the central ideas in the Catholic doctrine of salvation:
Adam and Eve, at their creation, were given a supernatural gift of grace (original holiness and justice/righteousness) by which they were able to love God supremely and obey him. However, they were tempted and fell into sin, rebelling against God, preferring their own ways to him. As a result of this, Adam and Eve entered into a state of mortal sin, which consisted of the guilt of their rebellion and consequent desert of eternal damnation, as well as a new fallen condition in which, without a new grace given, they would be unable to love God supremely and would forever prefer inferior goods to him. Their basic human constitution (consisting of natural human characteristics such as reason and will) were not destroyed by their fall, but their faculties were bent away from God and their will became inclined to sin. Since their will was not destroyed, they were still responsible for their choices, but they were so bent towards sin that they would never be able to be reconciled to God or turn back to him as their chief good (and nor would they be able to overcome the other effects of sin, such as alienation within themselves and with others and with the rest of creation) without new grace from God. Nothing in their human nature, without grace, could at all remedy this situation.
When Adam sinned, he lost his state of original justice and holiness and entered into a fallen condition, but his sin did not affect him alone. As father of the human race, his fallen condition was passed down from him to all his natural descendants. As a result, all humans are naturally conceived and born in a state deprived of original justice and holiness and subject to all the disorders this state naturally brings. Because of the fact that this fallen condition inevitably (without grace) inclines to sin and damnation, it came to be called the state of "original sin." In those (such as very young children) who are incapable of engaging in moral activity of their own (due to lack of ability to reason abstractly, etc.), this fallen condition does not result in personal sin and guilt immediately (because they are not capable of it), but it will inevitably lead to a personal state of mortal sin once an adequate capacity to reason and engage in moral actions develops (such as when children reach such a capable age). Thus, we can distinguish between "original sin" which is the condition that leads to sin and "personal sin" which inevitably results from it in those capable of such. As a result of the Fall, then, grace aside, all human beings who have reached the age of reason and personal moral actions are in a state of mortal sin that it is impossible they should emerge from and which will lead inevitably to eternal damnation (the worst part of which consists of the infinite loss of God and his blessedness and the fullness of misery that accompanies this).
This fallen condition can only be overcome by the supernatural grace of God, merited by the sacrifice and righteousness of Christ and applied by the Holy Spirit. God offers his grace to all men, and so all are without excuse for not turning back to God in reliance on that grace, and yet no one will ever have the will to turn back unless they are moved by grace to do so. When God's grace converts a soul (looking specifically at the soul of an adult), he applies actual grace to the will, turning it back to God so that the person comes to repent of his sins, love God above all else for who he is, and sincerely and fully receive Christ and his mercy. Thus, moved by grace, the convert leaves his state of mortal sin and enters into a state of grace, into a state of forgiveness of sins and holiness. While this transition occurs by means of cooperation between God's grace and the man's will, yet the entire transition, including man's very change of will, must be ultimately attributed to the grace of God, for man's good will is itself a result of grace and without grace man can do nothing. There are some to whom God gives grace only temporarily, without the gift of final perseverance, and so they only taste of Christ temporarily and do not ultimately attain to eternal salvation. But to God's elect, chosen from eternity, God gives the fullness of his grace, including the gift of final perseverance and the full fruition of eternal salvation in the enjoyment of God. God's grace also often works on infants, who are rescued from original sin (though the inclination to sin is not wholly removed from them, or from adults in this life) and restored to a state of justice, though the personal moral fruits of this will only appear later in life.
In the next life, the saved will be fully purified of all sin permanently, but in this life Christians still must struggle by the power of grace against their remaining inclination to sin, and they often fall into various sins. Sometimes God allows them to fall out of a state of grace entirely (mortal sins), and then moves them by his grace to restoration. Other times he allows them to fall into sin to a lesser degree, such that while they experience a sinful disorder it is not to the extent that it disrupts or destroys their overall relationship of love to God (venial sins). All sin by nature is in opposition to God and its natural fruit is alienation from God and misery, and yet not all sins are such as to remove one from a habitual state of grace. (To use an analogy, think of the difference between a fatal infection and a non-fatal infection. By its very nature, all infection tends to death, but not all infection actually infests the body in such a way as to bring destruction to the body overall and therefore bring death.)
God's grace works above and beyond his sacraments (such as his prevenient grace that moves the will to resort to the sacraments in the first place), and yet God's habitual communion with his people takes place (ordinarily) not in a condition of isolation from Christ's church but in communion with the rest of it and through the reception of the sacraments. For example, when a person is moved by grace to repent and turn to Christ, his reconciliation with Christ is liturgically enacted in his baptism (or in the Sacrament of Reconciliation if he has already been baptized). This is not to say that the grace of God is tied to the sacraments--for example, a person who has turned to God and desires baptism but is not able to receive baptism (perhaps he dies before this is possible, or he is kept from it by some external obstacle) is still saved by the grace and Spirit of God (this is usually called "the baptism of desire")--but that God's grace and his communion with man is ordinarily facilitated largely through the sacraments.
Along with the eternal consequences of sin and grace, such as eternal damnation and its forgiveness and the fundamental conversion of the soul, we must also take into account temporal consequences of sin and grace. Those who are forgiven of their sins and have entered a state of grace are not always necessarily freed from all the temporal consequences of their sins, and God's grace works not only to grant eternal salvation but also to wean man from sin through various trials and penances in this life (and also oftentimes in purgatory after death, which completes the purification of the regenerated soul so that he is fully fit for the full enjoyment of God).
Those who belong to the Augustinian and Thomist traditions within the Catholic Church go on to develop what they see as certain biblical and logical implications of the above doctrines: If the converted man's good will is entirely the result of God's grace, so that God works in us "both to will and to do" his good pleasure, then it follows that those who have come to God have done so because they have been given a special grace not given to those who refuse to come to Christ. And those who persevere to the end must likewise have been given a special grace that is not given to those who fail to persevere. Thus we must distinguish between the sufficient grace given to all men (which makes men inexcusable for not turning to Christ in reliance on his grace but which does not in itself turn man from sin to God) and the special efficacious grace which, of its own efficacy, actually turns men to God. From all eternity, God has chosen some out of the mass of fallen humanity to bring to eternal salvation. These are not chosen for any goodness they have over and above the rest, but solely by God's good pleasure and mercy. To these elect, God has chosen freely to give the efficacious graces that will lead not only to conversion but also to eternal salvation. To the rest, the reprobate, God has chosen to pass over them and not give them all the same graces (which is not unjust, for no one deserves these graces; they are an unmerited, gratuitous gift), and therefore they do not actually come to Christ, or, if they do temporarily, they do not persevere to the end. All of this follows from the fact that God's grace is the ultimate source of all good in man, and it also follows from the full sovereignty of God over all the creation, both fundamental doctrines of the Catholic faith.
The last paragraph alludes to a controversy between a couple of different schools of opinion in the Catholic Church regarding the doctrines of predestination and efficacious grace. One of the schools (and I'm defining "schools" broadly here) is the Augustinian/Thomistic school, whose view is described in the paragraph above. The other school is the Jesuit/Molinist school, whose views on predestination and efficacious grace are similar to those of the Arminian Protestant school. I talked a bit about the relationship between these two schools and how the Catholic Church has dealt with this controversy here:
The RC church is absolutely clear on the core of the gospel. Particularly, as part of this, the church affirms two things (among other things): 1. Free will. That is, the existence of true, voluntary choice, including in connection with salvation and damnation. 2. The absolute graciousness of salvation. Even the response of faith in the gospel is explicitly affirmed to be a work of grace in the heart. There is nothing we have that we have not received. (See, for example, here, and throughout this section of the Catechism.) The church allows both the Jesuit and the Augustinian positions on predestination and grace to be held by people in the church, and for people on both sides to argue with each other and support their own system, but they cannot deny these core elements of the gospel. The Jesuits think (wrongly) that the Augustinian position, taken to its logical conclusion, destroys free will. The Augustinians think (rightly) that the Jesuit position, taken to its logical conclusion, destroys the graciousness of salvation (and the supremacy and sovereignty of God, another point affirmed clearly by the RC church). Looking at things from the Augustinian point of view (that is, the right point of view), what the church has done is allow people who are confused on predestination and efficacious grace to continue to live in the church, but it does not allow them to affirm the full logical consequences of their position, thus showing care and charity for individuals while maintaining the graciousness of the gospel and the sovereignty of God. Looking at things from the Jesuit point of view (that is, the wrong point of view), the church has likewise shown charity and tender care by allowing those confused Augustinians to continue to live in the church while forbidding them to take their position to its logical conclusions, thus preserving the doctrine of the true voluntariness of human actions, including in connection with receiving or rejecting the gospel. If the RC church condemned the Jesuit view, the Jesuits would be forced to conclude, looking at things from their theological perspective, that the church had condemned voluntary human agency. If the RC church condemned the Augustinian view, the Augustinians would conclude that in rejecting these ideas logically bound up with and implied in the gospel, the church had rejected the gospel (and they would be right). So the church balances the purity of the gospel with charity towards individuals.
If you want to read more about the basic Catholic doctrine of salvation from official Catholic sources, see especially here, here, and here.
If you want to see something on the Thomist view of efficacious grace from a Thomist, see here and here. You can read more of the complete book here, or find it here. (Father Garrigou-Lagrange gives a good description of some of the history of the church's dealings with predestination, efficacious grace, etc., in these chapters and in his whole book as well.)
In this article, Jimmy Akin does a good job showing how much of the Calvinistic doctrines of grace can find a home in the Catholic Church.
Finally, in this lecture (don't miss where you can click to download the mp3s), Professor Lawrence Feingold does a good job discussing the controversy between the Thomists and the Molinists (though be aware that he writes from a Molinist point of view). The text of (I believe) the same talk can be found here. See here and here for a couple more good articles from Fr. John Harden on the controversy (he too seems to be a Molinist or close to it in his views). The Catholic Encyclopedia has an article on the controversy here.
ADDENDUM 7/28/15: I have the impression (new to the Catholic Church though I am) that a lot of Catholics do not have a clear awareness of the Thomist/Augustinian position within their church, and they sometimes describe what sounds something like the Molinist position as if it were THE official Catholic point of view. They condemn as "Calvinist heresy" ideas that Thomists/Augustinians hold as well. Related to this, I think a lot of Catholics, even Thomists/Augustinians, have a somewhat distorted view of what Calvinism is. They paint it as fatalistic, denying free will, etc., when Calvinism at its best is not that way I think (and I was a Calvinist for around eighteen years, so it's possible I might know something about it :-) ). There are certainly Calvinists who are that way, but that is not the mainstream Calvinism I came to know. I think we need to work on creating a more charitable, accurate understanding of Calvinism in the Catholic community (as Jimmy Akin has done in the article linked to above).
ADDENDUM 7/28/15: I thought I'd paste a couple of selections (with footnotes added into the text itself) from Fr. Hardon's summary (from here) of how the controversy has been decided (thus far) by the Holy See and thus what the current official attitude towards the various parties is:
Representatives for the two orders met before the papal commission in the Congregation De Auxiliis, so called because the auxiliary function of grace was under dispute. Three series of meetings were held between 1597 and 1607. In the first series, the decisions were highly critical of Molina and the Jesuits, urging the Pope to condemn their teaching. The final process was conducted under Paul V who dissolved the Congregation on September 5, 1607. His decision was to leave the final judgment to the Holy See while enjoining charity on the contending parties.
The De Auxiliis was unique in theological history. Each debate lasted several hours. At the first meeting, Alvarez (Dominican) and Valencia (Jesuit) disputed for a full four hours. The report for July 27, 1602, states that the disputation lasted uninterruptedly for seven hours on end. Sixty-eight distinct meetings took place under Clement VIII; at thirty-seven of these the theologians disputed, while the cardinals and consultors deliberated at the others. All told eighty-five congregations were held under Clement VIII and Paul V. What raised these discussions to a historic level was the personal assistance of the Pope at the various meetings, and the sincere effort made by the Holy See to settle the issue in one or the other direction, while finally deciding, in the words of Paul V, that “in treating of this question, neither side may condemn the position opposite to his own or charge it with any censure. Even more he desires that they abstain from using harsh epithets that betray animus towards one another.” [3--DS 1997.] . . .
When Benedict XIII in 1727 reconfirmed the privileges of the Order of Preachers, he forbade anyone to say that the doctrine of St. Thomas or his school was impugned by the condemnation of Jansenism. He than added that, “having discovered the mind of our predecessors, We do not wish either by our own or their praises conferred on the Thomistic school, which we approve and confirm by our repeated judgment, that there be any disparagement of the other Catholic schools which think differently from the same in explaining the efficacy of divine grace, and whose merits are also clear to the Holy See.” [6--Ibid.] He renewed the decrees of Paul V and forbade anyone “to brand with any mark of theological censure the schools that have different opinions” from the Thomistic position.
His successor, Benedict XIV, in 1748 came to the defense of the Molinists in a detailed statement to the Grand Inquisitor of Spain. It is the latest, authoritative declaration on the subject, which briefly summarizes the various schools of thought permissible in Catholic theology on the efficacy of grace.
You know there are manifold opinions in the schools on the famous questions about predestination and grace, and on the manner of reconciling human liberty with the omnipotence of God. The Thomists are said to be destroyers of human liberty and followers not only of Jansenism but of Calvinism. However, since they meet the charges with eminent satisfaction, and since their opinion has never been condemned by the Holy See, the Thomists carry on without hindrance in this matter, and it is not right for any ecclesiastical superior in the present state of affairs to force them to change their opinion.
The Augustinians are reported as the followers of du Bay and of Jansenism. They represent themselves as defenders of human liberty, and strenuously answer their critics. Since their opinion, too, has not been condemned by the Holy See, no effort should therefore be made to compel them to give up their theory.
The followers of Molina and Suarez are condemned by their adversaries as Semi-Pelagians. But the Roman Pontiffs have not passed judgment on the Molinist system, which they presently defend and may continue to do so. [7--DS 2564.]
ADDENDUM 10/15/15: See here for another post which discusses further the relationship of Catholicism to Calvinism and some of these other issues.
ADDENDUM 1/18/16: See the comment section in this article for further discussion of the toleration of both Molinists and Augustinians/Thomists in the Catholic Church. Let me here very briefly clarify that the Catholic Church has not formally approved either the Augustinian/Thomist nor the Molinist theological positions. It has simply refrained, until now, from passing judgment one way or the other on them. If the Church ever positively approved Molinism, that would be a serious problem for me, for it would logically imply an endorsement of the logical implications of that system (which I hold to be evidently contrary to the core of Christianity and to reason). As it is, Augustinians cannot usurp the Church's authority and take the initiative personally to formally condemn Molinism as heresy and eject Molinists from the Church (nor would we all necessarily wish to do so or think this the best or right course of action), but we are free to point out what we see as the truth that Molinism, taken to its logical conclusions, implies heresy and a denial of the core of the gospel, and to affirm that the Church's official theology, taken to its logical conclusions and applied to this matter, excludes these logical conclusions of Molinism.