The indefectibility of the Church is a gift from Christ to the Church by which she is preserved to the end of the age as the "institution of salvation". She can neither perish from the world nor depart from "her teaching, her constitution and her liturgy". (Ott, 296) This gift of indefectibility is essential to Christ's purpose in establishing His Church as the means of continuing His saving work to all the nations and peoples of the world until the end of the age. Regarding this purpose, Pope Leo XIII wrote, "What did Christ the Lord achieve by the foundation of the Church; what did He wish? This: He wished to delegate to the Church the same office and the same mandate which He had Himself received from the Father in order to continue them." (Satis cognitum, 4)
The indefectibility of the Church follows from the nature of the union of Christ with the Church, His Body. Concerning this union, St. Augustine writes:
Let us rejoice then and give thanks that we have become not only Christians, but Christ himself. Do you understand and grasp, brethren, God's grace toward us? Marvel and rejoice: we have become Christ. For if he is the head, we are the members; he and we together are the whole man. . . . The fullness of Christ then is the head and the members. But what does "head and members" mean? Christ and the Church.St. Gregory the Great (whose feast day is November 10) writes the following:
Our redeemer has shown himself to be one person with the holy Church whom he has taken to himself.
And St. Thomas Aquinas writes:
Head and members form as it were one and the same mystical person. (ST III Q.48 a.2)
It is because of this union of Christ with the Church that the Church is indefectible. St. Augustine shows this when he says:
The Church will totter when her foundation totters. But how shall Christ totter? ... as long as Christ does not totter, neither shall the Church totter in eternity." (Enarr. in Ps. 103, 2, 5)
Elsewhere, writing about Psalm 48:9 (which is Psalm 48:8 in Protestant Bibles) St. Augustine says:
Let not heretics insult, divided into parties, let them not exalt themselves who say, "Lo, here is Christ, or lo, there." (Matt 24:23) Whoso says, "Lo, here is Christ, or lo, there," invites to parties. Unity God promised. The kings are gathered together in one, not dissipated through schisms. But haply that city which has held the world, shall sometime be overthrown? Far be the thought! "God has founded it forever." If then God has founded it forever, why fearest thou lest the firmament should fall?"
And in his Sermon to Catechumens on the Creed (1:6), St. Augustine writes:
The same is the holy Church, the one Church, the true Church, the catholic Church, fighting against all heresies: fight, it can; be fought down, it cannot. As for heresies, they all went out of it, like unprofitable branches pruned from the vine: but itself abides in its root, in its Vine, in its charity.
In these quotations we see the indefectibility of the Church grounded in the Church's ontological union with Christ as His Mystical Body. Because the life of Christ is indefectible, and because the life of the Church is the life of Christ, therefore the Church is indefectible. Those who deny the indefectibility of the Church are denying that this union of Christ with His Church is anything more than extrinsic. They imply that Christ's Mystical Body can become corrupted such that He may abandon His Body and take on a different body. By their denial of the indefectibility of the Church they imply that Christ can abandon the Bride with which He is "one flesh" (Eph 5:31-32; Mt 19:6), and find a different bride. But such claims are contrary to the intimate and ontological union of Christ with His Body, which is also His Bride. In virtue of this union she can be neither defeated nor corrupted nor destroyed, since the risen Christ Himself can neither be defeated nor corrupted nor destroyed, and since His Spirit lives within her as her soul. (CCC 797, 809).
Understanding the indefectibility of the Church is essential for understanding what Christ did in establishing the Church and equipping her to fulfill His purpose. If the Church had not been given this gift, then each man would have been left to determine for himself what Christ's message was, by sifting through all the available historical evidence. The full and visible unity of all Christians would then be impossible, as each man would simply do what seemed right in his own eyes. That is why understanding indefectibility and its basis and necessity are essential for the reunion of all Christians. Only if indefectibility is true is there a means for full and visible unity. If the Church were not indefectible, every heresy would have an equal claim to orthodoxy with orthodoxy itself. But God cannot lie (Titus 1:2), and Christ has promised that the gates of hell would never prevail against His Church (Matthew 16:18), that He would be with her to the end of the age (Matt 28:20), that His Church is the pillar and ground of truth (1 Tim 3:15), and that His Holy Spirit would guide her into all truth (John 16:13). Therefore heresy has no equal claim to orthodoxy. Heresy is distinguished from orthodoxy precisely by what the indefectible Church has ruled.
The Catholic Encyclopedia article titled "The Church" has a very helpful section on the indefectibility of the Church. The following two paragraphs are excerpted from that article.
Among the prerogatives conferred on His Church by Christ is the gift of indefectibility. By this term is signified, not merely that the Church will persist to the end of time, but further, that it will preserve unimpaired its essential characteristics. The Church can never undergo any constitutional change which will make it, as a social organism, something different from what it was originally. It can never become corrupt in faith or in morals; nor can it ever lose the Apostolic hierarchy, or the sacraments through which Christ communicates grace to men. The gift of indefectibility is expressly promised to the Church by Christ, in the words in which He declares that the gates of hell shall not prevail against it. It is manifest that, could the storms which the Church encounters so shake it as to alter its essential characteristics and make it other than Christ intended it to be, the gates of hell, i.e. the powers of evil, would have prevailed. It is clear, too, that could the Church suffer substantial change, it would no longer be an instrument capable of accomplishing the work for which God called it in to being. He established it that it might be to all men the school of holiness. This it would cease to be if ever it could set up a false and corrupt moral standard. He established it to proclaim His revelation to the world, and charged it to warn all men that unless they accepted that message they must perish everlastingly. Could the Church, in defining the truths of revelation err in the smallest point, such a charge would be impossible. No body could enforce under such a penalty the acceptance of what might be erroneous. By the hierarchy and the sacraments, Christ, further, made the Church the depositary of the graces of the Passion. Were it to lose either of these, it could no longer dispense to men the treasures of grace.
The gift of indefectibility plainly does not guarantee each several part of the Church against heresy or apostasy. The promise is made to the corporate body. Individual Churches may become corrupt in morals, may fall into heresy, may even apostatize. Thus at the time of the Mohammedan conquests, whole populations renounced their faith; and the Church suffered similar losses in the sixteenth century. But the defection of isolated branches does not alter the character of the main stem. The society of Jesus Christ remains endowed with all the prerogatives bestowed on it by its Founder. Only to One particular Church is indefectibility assured, viz. to the See of Rome. To Peter, and in him to all his successors in the chief pastorate, Christ committed the task of confirming his brethren in the Faith (Luke 22:32); and thus, to the Roman Church, as Cyprian says, "faithlessness cannot gain access" (Epistle 54). The various bodies that have left the Church naturally deny its indefectibility. Their plea for separation rests in each case on the supposed fact that the main body of Christians has fallen so far from primitive truth, or from the purity of Christian morals, that the formation of a separate organization is not only desirable but necessary. Those who are called on to defend this plea endeavour in various ways to reconcile it with Christ's promise. Some, as seen above (VII), have recourse to the hypothesis of an indefectible invisible Church. The Right Rev. Charles Gore of Worcester, who may be regarded as the representative of high-class Anglicanism, prefers a different solution. In his controversy with Canon Richardson, he adopted the position that while the Church will never fail to teach the whole truth as revealed, yet "errors of addition" may exist universally in its current teaching (see Richardson, Catholic Claims, Appendix). Such an explanation deprives Christ's words of all their meaning. A Church which at any period might conceivably teach, as of faith, doctrines which form no part of the deposit could never deliver her message to the world as the message of God. Men could reasonably urge in regard to any doctrine that it might be an "error of addition".