This may be common knowledge, but can someone explain the process to me?
I doubt there’s any formal procedure.
“Ex Cathedra” means “from the chair”, and is when a declaration is made by the Pope, acting formally in his capacity as the successor of St Peter in exercising that authority given to him. In the 1950 apostolic constitution of Pope Pius XII Munificentissimus Deus, in which the dogma of The Assumption of the Blessed Virgin Mary was solemnly defined, the document ends with the following “formal” infallible assertion:
"44. For which reason, after we have poured forth prayers of supplication again and again to God, and have invoked the light of the Spirit of Truth, for the glory of Almighty God who has lavished his special affection upon the Virgin Mary, for the honor of her Son, the immortal King of the Ages and the Victor over sin and death, for the increase of the glory of that same august Mother, and for the joy and exultation of the entire Church; by the authority of our Lord Jesus Christ, of the Blessed Apostles Peter and Paul, and by our own authority, we pronounce, declare, and define it to be a divinely revealed dogma: that the Immaculate Mother of God, the ever Virgin Mary, having completed the course of her earthly life, was assumed body and soul into heavenly glory.
Hence if anyone, which God forbid, should dare willfully to deny or to call into doubt that which we have defined, let him know that he has fallen away completely from the divine and Catholic Faith.
In order that this, our definition of the bodily Assumption of the Virgin Mary into heaven may be brought to the attention of the universal Church, we desire that this, our Apostolic Letter, should stand for perpetual remembrance, commanding that written copies of it, or even printed copies, signed by the hand of any public notary and bearing the seal of a person constituted in ecclesiastical dignity, should be accorded by all men the same reception they would give to this present letter, were it tendered or shown.
It is forbidden to any man to change this, our declaration, pronouncement, and definition or, by rash attempt, to oppose and counter it. If any man should presume to make such an attempt, let him know that he will incur the wrath of Almighty God and of the Blessed Apostles Peter and Paul."
At Vatican II, Lumen Gentium 25 goes into further detail about how we are to understand infallibility, with a wider explanation including the Bishops:
“Although the individual bishops do not enjoy the prerogative of infallibility, they can nevertheless proclaim Christ’s doctrine infallibly. This is so, even when they are dispersed around the world, provided that while maintaining the bond of unity among themselves and with Peter’s successor, and while teaching authentically on a matter of faith or morals, they concur in a single viewpoint as the one which must be held conclusively.”
So a single idea or concept has to be defined fully, formally and conclusively without dispute as being a fundamental tenet of the Catholic faith, which must be held everywhere, always and universally if one is to define themselves as Catholic.