The term "sign of contradiction" refers to persons, organizations or teachings that, upon manifesting holiness, are subject to extreme opposition. The concept first appears in Luke 2:34, at our Lord's Presentation in the Temple. Simeon says to the Blessed Virgin Mary:
"Behold this child is set for the fall and rising of many in Israel, and for a sign that is spoken against (and a sword will pierce through your own soul also), that the thoughts of many hearts may be revealed."
Our Lord is the first and eternal sign of contradiction, and those who follow and imitate him often are opposed as He was. The Church is a sign of contradiction, being persecuted throughout its history. Likewise many saints, religious orders, and Catholic doctrines have become signs of contradiction when they were opposed by the enemies of Christ.
Today's feast day celebrates the Immaculate Conception of the Blessed Virgin Mary. From its promulgation by Blessed Pope Pius IX in 1854, this doctrine has been "a sign that is spoken against, that the thoughts of many hearts may be revealed." This teaching was the first to make explicit use of the power of papal infallibility, and thus many who opposed infallibility also rejected the Immaculate Conception. The Orthodox Churches (who generally accept the other infallible Marian doctrine, the Assumption of our Lady) reject this teaching because they do not believe in either Original Sin or the power of the Pope to define doctrine. Protestants do not accept the Immaculate Conception because they claim it is not found in Scripture. Modern secular humanism rejects this doctrine because it's unnecessary. What use is an Immaculate Conception if all people are sinless?