/u/TitusBluth gave a thorough answer to this in this answer to a question about the history of the papacy.

The short answer to the second half of your question is that it might be a little more cut-and-dried than you think; infallibility only applies when the pope "proclaims by a definitive act a doctrine pertaining to faith or morals," according to the Catechism. Has to be specifically invoked and to be pertinent only to doctrine or morality.

So the seeds of papal infallibility, if you will allow the expression, are certainly early, and reading the answer linked to in another place in this thread will provide a lot of information about that. I am here to talk about when papal infallibility was first explicitly held. The answer is late in the 13th century. The idea that the Pope was infallible in declarations concerning faith and morals, and that if he espoused heresy he was no longer a pope but an anti-pope, is first found in the writings of the Franciscans Pietro Olivi. Slightly obscure there, it comes to greater prominence when the more famous William of Ockham spilled a great deal of ink defending this idea of papal infallibility in his struggle with Pope John XXII.

Wait, what is that you say? The man contending with the pope was the one defending papal infallibility? Curiously enough, yes. You see, William of Ockham and Pietro Olivi were both Franciscans, that is to say members of the religious order founded by St. Francis of Assisi. St. Francis had very strict ideas about poverty, especially in what sense Christ and the Apostles practiced poverty and how Francis' own followers ought to live out. In 1279 Pope Nicholas III in his bull Exiit qui seminat basically agreed with the mainstream Franciscan interpretation, and stated that all property given to the Franciscan order would be owned by the Holy See, but that the Franciscans could have use of it. The Franciscans knew, however, that their ideas were still controversial and that any pope after Nicholas III could declare heretical what Nicholas III had allowed. Thus the idea of papal infallibility was made explicit, because Olivi thought this would protect the Franciscan ideal of Apostolic Poverty. Having already been declared licit by the pope, Olivi wanted to make clear that any subsequent pope that spoke against this would be speaking against an infallible statement, thus making himself a heretic and no longer a pope. Of course, this happened in 1323 when Pope John XXII forbade the Franciscan idea of Apostolic Poverty in his Papal Bull cum inter nonnullos, and it was this that made William of Ockham contend vigorously for the last 20 years of his life that John XXII was no pope at all.

So yes, papal infallibility has its roots earlier on, but it was first made explicit in the 13th century as a way of limiting the power of the papacy, not expanding it. Of course, Papal Infallibility as defined by Vatican I is somewhat different, perhaps we could say more nuanced, but that is the story of the first explicit formulation of papal infallibility by a theologian. Sorry I didn't give more background, but I need to be headed off to bed.

Source(s): I am ashamed to say I really have only read one book on this topic, namely Brian Tierney's The Origins of Papal Infallibility: 1150-1350. If anybody knows of a more recent work that would revise Tierney's narrative, please let me know.

Edit: I suppose I should nuance this at least a little bit. Tierney has a whole section at the beginning of his book arguing that papal infallibility as such can't be found in the canonical tradition of the 12th century (that is to say in Gratian's famous Decretum and its many commentators, such as Huguccio). Alfons Stickler in his review of Tierney's book (The Catholic Historical Review, Vol. 60, No. 3 (Oct., 1974), pp. 427-441) disagrees. I don't think myself qualified to adjudicate this dispute.