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"Conciliarism" refers to a movement in the medieval Catholic Church that sought to promote the authority of a council of bishops over that of the Pope; it was eventually condemned at the Fifth Lateran Council. In this case, the "conciliar folly" being referred to is the Second Vatican Council, which many so-called "traditionalist" Catholics reject on the grounds that the reforms it introduced were heretical, sacrilegious, or impious. Though of course the Second Vatican Council is in no way associated with the long-suppressed medieval conciliar movement, the author appears to be trying to make such a comparison in order to discredit it.
The Society of St. Pius the X is a Catholic organization formed in the wake of Vatican 2 specifically to resist those changes.