Outside the Church There is No Salvation?

The straight forward but constantly misinterpreted doctrine that "Extra Ecclesiam nulla salus": "Outside the Church there is no salvation", is the source of much confusion among many Catholics and non-Catholics alike. Does it condemn anyone not in communio sacris with the Church to damnation? To answer in the affirmative would place limits on God's grace and the salvific Incarnation and ministry of Jesus Christ. So, what exactly does this doctrine mean?

The Catholic Church teaches that a person may be saved outside the visible boundaries of the Church, (Lumen Gentium, sec. 19) provided the following conditions are met:

1. A person who, through no fault of his own, is ignorant of the truth that Christ established the Catholic Church and desires all men to obtain salvation through it (invincible ignorance).
2. A person lives according to the truth that he is capable of discerning.
3. A person responds to the graces that he has been given.

Or, re-formulated positively, the doctrine my be expressed as: "All salvation comes from Christ the Head through the Church which is his Body". According to the Catechism of the Catholic Church:

[846] "Basing itself on Scripture and Tradition, the Council teaches that the Church, a pilgrim now on earth, is necessary for salvation: the one Christ is the 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, 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."

[847] "This affirmation is not aimed at those who, through no fault of their own, do not know Christ and his Church: Those who, through no fault of their own, do not know the Gospel of Christ or his Church, but who nevertheless seek God with a sincere heart, and, moved by grace, try in their actions to do his will as they know it through the dictates of their conscience - those too may achieve eternal salvation."

[848] "Although in ways known to himself God can lead those who, through no fault of their own, are ignorant of the Gospel, to that faith without which it is impossible to please him, the Church still has the obligation and also the sacred right to evangelize all men."

A person enters the Church through baptism. Salvation is what God bestows upon those who seek God with a sincere heart, moved by grace, and attempt in their striving to do God’s will as they understand it compelled by their conscience.

Numerous Church Fathers, Doctors, popes and saints have articulated this doctrine:

[The Church] is the entrance to life; all others are thieves and robbers. On this account we are bound to avoid them… We hear it declared of the unbelieving and the blinded of this world that they shall not inherit the world of life which is to come… Resist them in defense of the only true and life giving faith, which the Church has received from the Apostles and imparted to her sons.

— St. Irenaeus

Nay, though they should suffer death for the confession of the Name, the guilt of such men is not removed even by their blood…No martyr can he be who is not in the Church.

— St. Cyprian

Abhor all heretics... heed not their fair speaking or their mock humility; for they are serpents, a 'brood of vipers.' Remember that, when Judas said 'Hail Rabbi,' the salutation was an act of betrayal. Do not be deceived by the kiss but beware of the venom. Abhor such men, therefore, and shun the blasphemers of the Holy Spirit, for whom there is no pardon. For what fellowship have you with men without hope. Let us confidently say to God regarding all heretics, 'Did I not hate, O Lord, those who hated Thee, and did I not pine away because of Your enemies?' For there is an enmity that is laudable, as it is written, 'I will put enmity between you and the woman, between your seed and her seed.' Friendship with the serpent produces enmity with God, and death. Let us shun those from whom God turns away.

— St. Cyril of Jerusalem

Let no man deceive himself. Outside this house, that is, outside the Church no one is saved.

— Origen

No man can find salvation except in the Catholic Church. Outside the Catholic Church one can have everything except salvation. One can have honor, one can have the sacraments, one can sing alleluia, one can answer amen, one can have faith in the name of the Father and of the Son and of the Holy Ghost, and preach it too, but never can one find salvation except in the Catholic Church.

— St. Augustine

There is no entering into salvation outside the Church, just as in the time of the deluge there was none outside the ark, which denotes the Church.

— St. Thomas Aquinas

The First Vatican Council decreed that it is the Magisterium's exclusive duty to determine and explain the meaning of Sacred Tradition, including "Outside the Church there is no salvation." See Fr. William Most's answer discussing "Is There Salvation Outside the Church?" (EWTN).

[Update: 11/4/2015] As St. Camillus correctly notes, (see comments) Extra Ecclesiam Nulla Salus has been infallibly defined three times by popes and councils, and piously believed by the faithful in every age of the Church's existence:

There is but one universal Church of the faithful, outside which no one at all is saved.

— Pope Innocent III, Fourth Lateran Council, 1215.

We declare, say, define, and pronounce that it is absolutely necessary for the salvation of every human creature to be subject to the Roman Pontiff.

— Pope Boniface VIII, the Bull Unam Sanctam, 1302.

The most Holy Roman Church firmly believes, professes and preaches that none of those existing outside the Catholic Church, not only pagans, but also Jews and heretics and schismatics, can have a share in life eternal; but that they will go into the eternal fire which was prepared for the devil and his angels, unless before death they are joined with Her; and that so important is the unity of this ecclesiastical body that only those remaining within this unity can profit by the sacraments of the Church unto salvation, and they alone can receive an eternal recompense for their fasts, their almsgivings, their other works of Christian piety and the duties of a Christian soldier. No one, let his almsgiving be as great as it may, no one, even if he pour out his blood for the Name of Christ, can be saved, unless he remain within the bosom and the unity of the Catholic Church.

— Pope Eugene IV, the Bull Cantate Domino, 1441.

Prayer for the Church

Remember O Lord, Thy Holy Catholic and Apostolic Church which Thou hast purchased with Thy precious blood. Confirm and strengthen it, enlarge and multiply it, keep it in peace, and preserve it unconquerable by the gates of hell forever. Heal the schisms of the churches, quench the ragings of the heathen, speedily undo and root out the growths of heresies, and bring them to naught by the power of Thy Holy Spirit. Make it faithful always, in imitation of You. Amen.