We finally see the actual Papal Mainframe in the latest Christmas Special, "the Time of the Doctor." It is an incredibly massive computer and the data-base it contains, much like "the Library." It save-guards the security of information "in this life and the next" across political lines for the Known Universe at some point, it is the organization encompassing the "Church of the Silence" and all of it's sects, including Madame Gravorian's branch that used time-travel and assassins, including an unrememberable T.A.R.D.I.S. bomber and a hybrid sociopath raised from childhood, to attempt to prevent the Doctor from ever reaching Trensalore.
The Mainframe itself seems to be the big blocky ship that can project massive holograms of the Mother Superioress' (the head-nun-in-charge, sort of a pope, a tough woman the Doctor has a history with, something of a sociopath ) head, though that ship, massive as it is, is never actually said to be the whole of the actual mainframe Itself.
It turns out that we have been seeing many reference to things that are Papal Mainframe-related for quite some time.