the "invisible church" is a concept used by Protestants to reconcile their rejection of the Catholic Church with the Scriptures (specifically the promise made by Christ that the Church would not fall).
the idea is that when Christ spoke about the Church he was in fact speaking about the "invisible Church". ie : the "real" one, formed by the Saints and all those who have been saved by God's (predestined) Grace since the beginning of the world.
Only God know who is part of this Church and who is not.
being a member of the "visible church" doesn't make you automatically a member of the invisible one.
The invisible Church is God's Church. It will not fall, and will not fail.
the visible(s) church(es) is "man made", and, as such, not only it can fall, but it probably will. repeatedly.
hence the need for a Reformation.
Augustine is often quoted as the source of this doctrine. While i don't think he ever conceived this specific concept (which arose much later), it's true that, as a student of Plato, he quite often used these kind of dialectical oppositions visible/invisible - phenomenal/noumenal - earthly/heavenly.