Indefectibility is an attribute of the Catholic Church. It means that the Church will endure until the end of time. Nothing can destroy it. It is based on Jesus saying that the gates of hell would not prevail against the Church (Matthew 16:18) and, “I am with you always even unto the end of time.” (Matthew 28:20)
Though indestructible, the Church has been persecuted through the ages. The Roman Empire persecuted the Church for three hundred years. The Church endured two schisms (Eastern and Western), the Reformation, the German Kulturkampf, Communist oppression, and much more. Even the Borgia and Medici scandals, and the recent clergy sex scandals in the USA, have hurt, but have not destroyed, the Catholic Church. Internal and external opposition have been present from day one, but the Church substantially continues to exist and thrive. With over one billion members around the world, a two-thousand-year-old church speaks for herself.
Indefectibility does not mean, however, that individual members do not need or cannot change or improve themselves. Every pope, bishop, priest, deacon, and layperson is called to do and be better every day of their lives. The Church as a whole is the bride of Christ and is the Mystical Body of Christ. It is in her individual human members that one finds vulnerability, weakness, imperfection, and even sinfulness. The Church as a whole cannot sin, but her members can all the way from the top on down.
The idea of the Church enduring until the end of time does not define the size, either. Saint Ambrose said ubi Petrus, ibi ecclesia meaning, “Where there is Peter (the pope), there is the Church.” As long as there is a pope, there is a Church.