First of all, if you want to read up on the basics of the Catholic doctrine of salvation from official Catholic sources (other than the Bible), these three sources are a good place to start: 1. The section on "grace and justification" in the Catechism of the Catholic Church. 2. Session V and Session VI of the Council of Trent, which deal with the topics of original sin and justification. (Only the first section of Session V, "Decree Concerning Original Sin," is relevant, the other part dealing with other matters.) 3. The Canons of the Council of Orange, which are directed against the heresy of Semipelagianism.
Now here is a statement of my own summing up some key points 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 who are never converted to Christ by God's grace. 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).
Eternal life is granted as a reward for good works, but those who are saved will have nothing to boast about, because all their works are nothing other than the work of God's grace, the fruit of Christ's righteousness applied to their lives by the Holy Spirit. Those who end up damned will have no one to blame but themselves, for they freely rejected God's offer of salvation. Damnation is ultimately to be attributed to the fallen free will of man, while salvation is ultimately to be attributed to the grace of God.
Salvation includes not just the forgiveness of sins and sanctification, but also adoption, whereby we become children of God in a special and supernatural way through our union with Christ, the one Son of God. By grace, we become co-heirs with Christ of the glory of God, just as he is the heir of his Father by nature. The ultimate fruit of salvation is to share in the life and love of the Blessed Trinity for all eternity.
The Catholic doctrine of salvation is a middle way that opposes two opposite errors. On the one hand it avoids the error of the Pelagians and Semipelagians, who attribute the saved man's good will (and thus his righteousness) ultimately to man himself and the use of his free will. Catholic faith affirms, on the contrary, that the ultimate source of the goodness of the regenerate is God's grace. On the other hand, Catholic faith avoids the error of those (including some, but not all, Protestants) who say that since grace is the source of salvation, there is no requirement for man to use his will to cooperate with grace or to strive to do good works, or that the saved have no good works that are truly pleasing to God. Catholic faith affirms, on the contrary, that while the grace merited for us through the passion and righteousness of Christ is the only ultimate source of salvation, yet God's grace works in us by moving our wills to cooperate with his grace so that we are made holy and do holy works, and that this inward holiness and these holy works, produced in us by grace, are truly pleasing to God and fit to be rewarded with eternal life.
Two quotations from the Apostle Paul in the New Testament well sum up both sides of the grace/works equation and how they relate to each other: 1. Ephesians 2:8-10: "For by grace are ye saved through faith; and that not of yourselves: it is the gift of God: Not of works, lest any man should boast. For we are his workmanship, created in Christ Jesus unto good works, which God hath before ordained that we should walk in them." 2. Philippians 2:12-13: "Wherefore, my beloved, as ye have always obeyed, not as in my presence only, but now much more in my absence, work out your own salvation with fear and trembling. For it is God which worketh in you both to will and to do of his good pleasure."
Some readers might be interested in following up on some of this with more specifics. If you are interested in getting a better idea of how the Catholic ideas of penance, purgatory, and indulgences fit into this overall scheme, see here. If you are interested in seeing some thoughts on how all of this relates to concepts like predestination and efficacious grace, see here and here. If you are interested in reading more about the nature of justification, see here, here, here, and here.