What is the Catholic method of salvation?

The Catechism is quite clear. Faith is absolutely necessary.

161 Believing in Jesus Christ and in the One who sent him for our salvation is necessary for obtaining that salvation.⁴² 'Since "without faith it is impossible to please (God)" and to attain to the fellowship of his sons, therefore without faith no one has ever attained justification, nor will anyone obtain eternal life "But he who endures to the end".'⁴³

The Catholic Church subscribes to the historic creeds, including the Athanasian Creed (see CCC192)

42 Cf. Mk 16:16; Jn 3:36, 6:40 et al.
43 Dei Filius 3; cf Mt 10:22, 24:13; Heb 11:6; Council of Trent

See also the related questions about works flowing from faith and sanctifying grace. Faith comes first. Everything else comes from that.

I am going to explain the Catholic doctrine in terms that a Protestant can understand. I would not usually answer a question best directed to Catholics (as I am a Protestant) but I find there is an extreme different understanding of the terms used. It is necessary therefore to provide translations for the common words that each uses so that the almost opposite meanings can be understood and turned back into ones own frames of reference.

Salvation to a Catholic is a process of sanctification not an instantaneous 'justification by faith', therefore although faith is required and important it does not carry the same meaning. Faith to a Catholic is therefore restricted to something within the state of grace, which makes a believers duty grow as they understand more accurately the truth in Christ. Faith is similar to protestant faith in this sense but does not have this prime direction towards a justification apart from works.

Entrance into a state of salvation While evangelicals generally believe a person is born again, or made alive, by faith in Christ, Catholics understand a person to enter into a similar state of salvation by Baptism into the Roman Catholic church

Justification Protestants actually put this first and they mean that a sinner is declared fully righteous before God as though they had obeyed all of God's laws perfectly due to the righteousness of Christ being charged, or imputed to them. This is an event, the result being entrance into the church, or new birth and sanctification.

Catholics do not believe in justification in this sense. To them justification is a lifelong process but when one is first justified, it is just the first moment of sanctification. In other words in Protestant terms there is no justification, only sanctification and if you finally die in a justified state, your are just and will enter heaven. However, as mentioned, the word justification can be used as the first moment of sanctification, whereby through external baptism into the church the sinful nature is weakened to such a degree that holiness is first implanted into the soul. It is the point where original sin is removed however there is no perfection granted to a sinner prior to actual holiness, rather the actual holiness infused into the soul is the thing that justifies you, i.e your sanctification or holiness and works. In other words, in Protestant terms you are saved by your own holiness, or works of the law, but these come from something Catholics also call grace. Therefore under a Catholic view you still say in a different sense that you are 'only saved by grace' and 'faith'. Protestants will understand that Catholic 'grace' is a confusing terms for what Protestants call works.

Continued Sanctification Catholics believe that you have minor and major sins, they are called venial and mortal. Basically venial sin is just sin for most evangelicals in its consequence.. In other words venial sins do not destroy your sate of grace. However they still stain the soul and you might need some purgatory if you have not repented and confessed those sins to a priest.

Mortal sins actually take you out of the state of grace, or sanctification. If you die under the guilt of an unconfessed mortal sin, you go to hell. Most evangelicals do not have a concept that equals mortal sin, however arminian theologians believe you can loose your faith, which in that sense is the only mortal sin (for those who believe this is a possibility). Many, if not most, evangelicals think a believer does not have the power to fully loose their faith or commit curtains sins like hating the church, becoming a mass murderer, teaching false doctrines, etc.

Summary Most evangelicals believe they are justified prior to sanctification. While still enemies of God and without any holiness they are declared holy. Then afterwards they are protected by that justification, through sanctification, all the way to glory. It is all free grace and not derived from works, though works will automatically accompany this great freedom. Catholics, on the other hand, feel there is no such thing as justification apart from your continued works. This is all silly nonsense talk. Rather, sanctification is all that matters and it starts with your baptism into the Roman Catholic church body and then is maintained by your effort, works and confession. This sanctification justifying you is done 'in cooperation' with something they also call grace. Of course by grace they mean something quite different from a Protestant.

I know this is all complicated but I hope I have brought some light to your very good question. The two views all hinge on the difference of justification in a 'moment' versus a long period of obedience.

Note: It might be worth noting to understand the full practical outcome that Catholic teaching does not consider anyone who willingly and knowingly rejects the authority of the Pope to be in a state of grace, as this is obviously a so called mortal sin. Therefore, the faith that Protestants have and which justifies them is viewed as false and basically a lie invented by heretics under Catholic dogma.

References:

All references are taken from a large classical Roman Catholic source, originally written to train Jesuit teachers in three volumes entitled 'OUTLINES OF DOGMATIC THEOLOGY BY SYLVESTER JOSEPH HUNTER, OF THE SOCIETY OF JESUS'.

Sylvester Joseph Hunter (Catholic Encyclopedia)

Baptism

In general once you are Baptised in the Catholic church you are saved:

The Effects of Baptism.—We learn from the testimony of Holy Scripture and from the other channels of tradition, that the Sacrament of Baptism, duly received, works many excellent effects in the soul; and first, it does away with all sin, both original and actual. (nn. 500, 596.) The Apostles baptized men for the remission of their sins (Acts 2:38); St. Paul was bidden to arise and be baptized and wash away his sins (Acts 22:16); the Corinthians are reminded of the sin in which they had had part, but are reminded also that they were washed, were sanctified, were justified. (1 Cor. 6:9–11.) These texts are in accord with the declaration of the Council of Trent that, by the grace of our Lord which is conferred in Baptism, the guilt of original sin is remitted, and all that has the true and proper character of sin is removed. (Sess. 5, can. 5; Denz. 674.) Further, the Council, in the same place, defines that in the newly baptized there is nothing to hinder their entry into Heaven; or, in other words, that the Sacrament releases from all temporal punishment due to forgiven sin. (n. 829.) That this doctrine was held by the Church in early days is proved by the distinction which she made, in the administration of her penitential discipline, between pre-baptismal and post-baptismal sin: penance was enjoined for the latter, but not for the former: a candidate for Baptism was exercised in works of penance, but, when once he was baptized, the past was forgotten. Another effect of the Baptism of water is that a character is impressed upon the soul, as we have seen (n. 671); and lastly, the baptized person becomes an adopted son of God, a member of Christ (Galat. 3:27; 1 Cor. 6:15), and is joined to the Church. (Acts 2:41.) (Outlines of Dogmatic Theology, Vol. 3, pp. 222–224)

It should be noted to accomodate the person who believes in Christ and has not yet been baptised 'within the Catholic church' but genuinly wants to, is still protected but of course this time must be very short and rare exceptions, if it is a genuine desire.

The Necessity of Baptism.—The Council of Trent (Sess. 6, cap. 4; Denz. 678) declares that since the promulgation of the Gospel, justification (n. 626) cannot be attained without Baptism of water or the desire of it, according to the words, “Unless a man be born again of water and the Holy Ghost, he cannot enter the Kingdom of God.” (St. John 3:5.) The words of this text are perfectly general, and the constant tradition of the Church teaches that they embrace all human beings, of whatever age. Baptism in some form is therefore the necessary means of salvation. Baptism of Desire.—But this necessity is not strictly absolute, in the sense in which the habit of grace (n. 643) is necessary to salvation; for, as the Council suggests, when the Baptism of water cannot be had, its want may sometimes be supplied. 'Outlines of Dogmatic Theology, p. 224)

Justification

Catholics treat justification as the sanctification which generall coincinced with Baptism (or the sincere desire to be baptised in teh Catholic church as a result of believing in the gospel).

The Remission of Sin.—It is the doctrine of the Catholic Church that when God receives a sinner to His friendship, all sin is altogether forgiven and becomes as if it had never been: the Lutherans on the other hand maintain that the sin and its guilt still continue in existence, but that God deals with the man as though he had no sin. The point is defined by the Council of Trent when it condemns all who deny that “by the grace of Jesus Christ conferred in Baptism the guilt of original sin is remitted,” or who assert that this guilt is “merely not imputed.” (Sess. 5, can. 5; Denz. 674.) The Council here speaks of original sin, which is remitted in Baptism along with all actual sin, if the person have any; but what is said applies no less to the remission of post-baptismal sin, which is the work of the Sacrament of Penance. It is not meant that, historically, the forgiven sin has no longer been committed, for God does not alter the past; but that morally, the sin is as if it never had been, for there is no longer in the soul anything that excites the wrath of God, and the stain arising from the absence of sanctifying grace is no longer there. We proceed to give the proof of this doctrine from Scripture. (Outlines of Dogmatic Theology,pp. 127–128)

Sactification

Catholics imagine sanctification, or salvation to be a result of God 'infusing goodness into us' by faith. This is decpetively similiar to the Proetsant view of sanctification however we must remember that Justification is just the first part of that infusion and that if this state is lost through mortal sin, so is the justification.

Infusion of Virtues.—It is the certain doctrine of theologians that some habits of virtue are infused into the soul when a sinner is justified, and some believe that the doctrine is absolutely of faith, at least as regards the theological virtues. This is the sense in which the Fathers understand the phrase of Scripture, that God will take away the heart of stone and give a heart of flesh (Ezech. 11:19), especially as it is added that this shall be done that men may walk in the commandments of God: the inward renewal therefore of which we have spoken (n. 636) must be some permanent change for good wrought in the heart by God: in other words, an infusion of virtue. The same meaning is expressed when Jeremias declares that God will write His law upon the hearts of His people. (Jerem. 31:33.) If any virtues be infused, it would seem that those at least must be included which are most noble and most necessary, that is to say, the theological virtues, as the Council of Trent teaches. (Sess. 6, cap. 7; Denz. 682.) In regard to the moral virtues there is more room for doubt, but the common teaching is that these also are infused. All hold that the habit of charity and habitual grace come to the soul together, and are lost at the same time, when mortal sin is committed: and so closely are they akin that Scotus, Bellarmine, and others hold that they are really the same: but the contrary opinion is more commonly followed, on the authority of the Thomist school and Suarez. The habits of faith and hope are, it is believed, not lost except when a sin is committed which is directly opposed to these virtues. The considerations on which these conclusions are based must be omitted as too subtle. (Outlines of Dogmatic Theology (Third Edition., Vol. 3, pp. 148–149)

Meritorious Good Works

Ultimately by having a decpetively subtle way of joining works of free will with grace, a concept of 'merits' results effectively meaning that one can 'derserve more grace from initail grace' which is effectively destroying the meaning of the word grace under the protetsant definiton. (However some forms of Arminina thinking are not that much different to this). Good works in this sense are at leat excluded from the initial moment of sanctification, like Protetsants, it is the maintetance of being in the state of grace, after Baptism and faith in the Catholic doctrine of the gospel, where meritorious good works is intended.

What is merited.—We have shown (n. 592) that it is impossible for man to merit, even congruously, the first actual grace; which in fact follows from what we have said, that there is no merit in works not done under grace. With the aid of grace, a sinner may merit further actual graces with congruous merit (n. 604); and of course, the just can do the same; especially they can merit congruously the grace of final perseverance. Final perseverance cannot be merited condignly (n. 599); but the objects of this merit are what are enumerated in the canon of the Council of Trent, lately quoted. (n. 653.) It will be seen that the texts of Scripture by which we establish the existence of merit extend to these objects. (Outlines of Dogmatic Theology,Vol. 3, pp. 159–160)

The key: Initial Sanctification or Inward Rewnewal

To ties up thee various toughts they key difference between Catholc and Protestant views is back on the topic of Justification. Catholic dogma asserts the the justification is caused by actual grace making the soul actually holy and just, not a justification for someone who is still dead in sins.

Inward Renewal.—It is a peculiarly Calvinistic error, that justification consists in the remission of sin alone, without renewal of the inward man. The Council of Trent condemns it in a form which embodies the proof from Scripture of the opposed Catholic doctrine. “If any one say that men are justified by the imputation of the justice of Christ alone, or by the remission of sins alone, apart from grace and charity which is poured forth in their hearts by the Holy Ghost, and is inherent in them; or again that the grace which justifies us is no more than the favour of God, let him be anathema.” (n. 494.) This doctrine is taught by St. Paul from whom the Council took the phrase concerning the action of the Holy Ghost (Romans 5:5); it is fully set forth in the Epistle to St. Titus (3:5), where it is said that God saved us by the laver of regeneration and renovation of the Holy Ghost, whom He poured forth upon us abundantly through Jesus Christ our Saviour: and many other passages are found where justification is described as new birth, by which one dead in sins receives the gift of life; in the words of the Prophet, not merely is the heart of stone taken away, but the heart of flesh is given. (Ezech. 11:19.) By this new birth, the sinner becomes truly just, participating in the justice of God. “The one formal cause of justification is the justice of God; not that by which He is just, but by which He makes us just: by the receipt of which gift we are renewed in the spirit of our mind and are not merely reputed just, but are named and actually are just, when we receive justice into ourselves, each according to his measure, which the Holy Ghost imparts to each as He pleases, and according to the disposition of each one and his co-operation.” Such are the terms in which the Council of Trent (Sess. 6, cap. 7; Denz. 681) opposes that doctrine of of “imputed righteousness,” which in one or another of its many varieties finds favour with the followers of Luther and Calvin. The teaching of the Council by no means denies that the justice of Christ has merited justification for us: it merely says that our justice is something different from the justice of our Redeemer. (Outlines of Dogmatic Theology, Vol. 3, pp. 129–131).

Note: The author does not attempt to explain that Calvinism does not exlude renewal of the inner man, but only exludes it from our justifcation. The author is right is understanding that justification comes prior to that renewal in theological order, as it must occur while the soul is still dead in sin. Then a moment after, the renewal occurs as a result of the justification under the Protestant doctrine.

According to What Catholics Really Believe: 52 Answers to Common Misconception About the Catholic Faith by Karl Keating (Ignatius Press):

The Catholic Church teaches we earn salvation by good works

The Catholic Church has never taught such a doctrine. In fact, it has constantly condemned the notion that we can earn salvation. Only by God's grace, completely unmerited by works, is one saved. The Church teaches that it's God grace from beginning to end which justifies, sanctifies, and saves us. As Paul explains in Philippians 2:13, "God is the one who, for his good purpose, works in you both to desire and to work."

Notice that Paul's words presuppose that the faithful Christian is not just desiring to be righteous, but is actively working toward it. This is the second half of the justification equation, and Protestants either miss or ignore it." [italics in original. p100.]

Another part of the book:

The Church teaches that as long as Catholics go to Mass and confess any serious sin, they will go to heaven.

It isn't enough to believe the Catholic faith--you also have to live it. In fact. how you live demonstrates what you believe. If you have what is termed "saving faith," it will manifest itself in a holy, obedient life. If your faith is nothing more than a list of propositions to which you give mental assent, you have only intellectual faith, the kind James says is insufficient for salvation (Jas 2:24)." [italics in original.] p21.

I am not a Catholic, and I found this book (What Catholics Really Believe: Answers to Common Misconceptions About the Faith) highly informative. You can get a copy from Barnes and Noble.