"No one can find salvation except in the Catholic Church. Outside the Church you can find everything except salvation. You can have dignities, you can have Sacraments, you can sing "alleluia", answer "Amen", have faith in the name of the Father, the Son, and the Holy Ghost, and preach it too; But never can you find salvation except in the Catholic Church." - St. Augustine of Hippo
"There is indeed one universal church of the faithful, outside of which nobody at all is saved, in which Jesus Christ is both priest and sacrifice." - 4th Lateran Council. 1215 C.E.
"Urged by faith, we are obliged to believe and to maintain that the Church is one, holy, catholic, and also apostolic. We believe in her firmly and we confess with simplicity that outside of her there is neither salvation nor the remission of sins." - Unam Sanctam. 1302 C.E.
"This holy Council first of all turns its attention to the Catholic faithful. Basing itself on scripture and tradition, it teaches that the Church, a pilgrim now on earth, is necessary for salvation: the one Christ is mediator and the way of salvation; he is present to us in his body which is the Church. He himself explicitly asserted the necessity of faith and baptism (cf. Mk. 16:16; John 3:5), and thereby affirmed at the same time the necessity of the Church which men enter through baptism as through a door. Hence they could not be saved who, knowing that the Catholic Church was founded as necessary by God through Christ, would refuse either to enter it, or to remain in it" - Vatican II "Lumen Gentium"
A Traditionalist, A Modernist, and A Pelagian Analysis
Catholic doctrine hasn't changed after Vatican II despite what you may have been told by your local modernists. When the Church says that there is no salvation outside the Church, it is important to note that if you have a Trinitarian baptism, you have been "saved" and are within the Church. But the Magesterium also teaches that those who are invincibly ignorant of Catholicism are not to be held accountable of it, and that "somehow" (Key word) people outside the visible boundaries of the Roman Church are able to be saved as well. Thus it was taught to young Irish immigrants that their Anglican neighbours were still loved by God because they didn't mean to commit the sin of schism. That is the acceptable doctrine of invincible ignorance. But a Traditionalist would probably say that if you've read this blog you are no longer invincibly ignorant (sorry for damning my Protestant readers, in the words of the Baptists: "Turn or Burn"). I've never heard an orthodox Catholic answer to the question: what if someone dies in repentance of their sin but refuses still to acknowledge the Catholic Church as valid and does not desire her absolution even though they know what she claims to be. I'm assuming the answer is that they are not saved.
Then another Catholic might say God "Freely accepts those who do what is right and fear him" (To quote St. Peter in his speech to Cornelius). They might say that Jews and even "godly pagans" are saved because they are doing good works. The idea that those who do good works outside of Christ are rewarded with Heaven is the Pelagian heresy. Now in this view's defense they might say that it is only semi-pelagianism in that all good works of humans come from grace. But I would quote Hebrews to say "without faith it is impossible to please him" and quote Romans to say "how can they believe in Him whom they have not heard of". So I don't think ordinarily those outside Christendom are saved, but who knows, it's up to God.
Finally you will find the modernist, even at the highest levels of our Church. For example, take the former president of the Catholic bishops of England and Wales, Cardinal Cormac Murphy-O'Connor who is a universalist. And his universalism isn't even based on scripture, tradition, or reason, no he's with Wesley on this in giving credence to emotions. He says he believes all will be saved because it will make him feel better in Heaven.
So there you have it, a cross-section of the whore that is our Church, and yet our mother (To quote St. Augustine).
"And this is the boldness we have in him, that if we ask anything according to his will, he hears us. And if we know that he hears us in whatever we ask, we know that we have obtained the requests made of him. If you see your brother or sister* committing what is not a mortal sin, you will ask, and God will give life to such a one—to those whose sin is not mortal. There is sin that is mortal; I do not say that you should pray about that. All wrongdoing is sin, but there is sin that is not mortal. We know that those who are born of God do not sin, but the one who was born of God protects them, and the evil one does not touch them. " - 1 John 5:14-18
The list of mortal sins is traditionally the list Jesus gives to the rich young ruler when he says "do this and live" in St. Luke's gospel. But Catholic moral theology holds that for a mortal sin to occur, it has to be a grave matter, with full knowledge of it's sinfulness, and full assent of the will.
So in regards to a question someone asked, yes if you are a baptized Protestant and have sinned gravely and willfully (I know I do this weekly), then according to Catholic teaching you have killed God's grace in your soul, and St. John the Divine says that people shouldn't even pray for you (though I'd pray for you anyway).
So seek priestly absolution (Jn 20:23), or persist in arrogance and heresy. Those are the mean-spirited, harsh, and unpastoral words I am obligated to say by Canon Law.
May God have mercy on our souls.