A friend of mine asked me today:
"What's 'infallibility'? I can't find the word in the Bible anywhere and I don't know what it means. Is it something that the Catholic Church just made up?"
Before I addressed the question, it was important to point out that there are many words used by Catholics and by non-Catholic Christians alike that do not appear or are mentioned in the Bible (Sacred Scripture). For example: the words "trinity", "pope", "incarnation", or "Eucharist" don't appear in the Bible yet as Christians we believe in the triune nature of God (the trinity), we as Catholics believe in the apostolic authority of the Church and that Jesus established a visible Church with St. Peter as its head (Matthew 16:18), we believe that Jesus is God in the flesh (the incarnation, John 10:30), and we partake in the sacrament of the Eucharist as Christ instructed us to at the Last Supper (Luke 22:19).
While certain words commonly used by Christians are not found in the Bible, they're used (i.e. coined) because they allow us to describe doctrines and concepts revealed to us in scripture. Basically, the word itself might not be there in the Bible, but the framework and evidence for it is.
So what's "infallibility"? In short, "infallibility" refers to the Church's ability to teach on matters of faith and morals infallibly. To teach infallibly means to teach with the guarantee that there is no error in the teaching and that it has not been taught in error.
Christ himself instructed the disciples to go out and teach and that He would be with them until the end of time (Matthew 28:19-20). Jesus would send the disciples His spirit to be with them and guide them in their teaching:
"But when he, the Spirit of truth, is come, he will teach you all truth. For he shall not speak of himself; but what things soever he shall hear, he shall speak; and the things that are to come, he shall shew you." - John 16:13
Christ bestowed upon St. Peter the authority to lead the visible church and the authority to "bind" and "loose" (Matthew 16:19) which refers to teaching authority.
"He that heareth you, heareth me; and he that despiseth you, despiseth me; and he that despiseth me, despiseth him that sent me." - Luke 10:16
Christ also, as we read in John 21:15-17, instructed St. Peter to feed His lambs/sheep. In other words, Christ entrusted St. Peter with the responsibility to guide all in Christ as a shepherd does his flock.
The Church is known as the "pillar and bulwark of truth" (1 Timothy 3:15) because the Holy Spirit guides it, and when the Church teaches infallibly it does so because the Holy Spirit has enabled it to. The teaching authority of the Catholic Church is known as the "Magisterium", taken from the Latin word "magister" which means "teacher" and this responsibility to teach is emphasised in Christ's instruction to the disciples to baptise and teach all nations as we read in Matthew 28:19-20.
The Church teaches infallibly when the pope declares an infallible teaching. When the pope does this, they are known to be speaking "ex cathedra" which means that they are speaking from the chair of St. Peter again referring to the authority given to St. Peter by Christ and the ability to "bind and loose". This does not, however, mean that the pope himself is infallible. The pope is a just a man; an ordinary human like you and me, only he has an extraordinary role as a successor of St. Peter and charged with the responsibility of leading the visible church as St. Peter did from around 33AD. Fundamentalists often make the mistake that the word "infallibility" refers to the pope himself and not to what the pope is charged with, that is to "bind and loose".
A particular instance of this "speaking from the chair of St. Peter" was exemplified at the Council of Chalcedon (circa 451AD) when Pope Leo and the bishops met to speak against Monophysitism, a belief that dictated that Christ only had one nature or a "half-half" nature (as opposed to the two distinct natures of Christ that we believe in known as the Hypostatic Union: fully human and fully divine; fully man and fully God). When Pope Leo presented his tome, it was declared:
"This is the faith of the fathers! This is the faith of the Apostles! So we all believe! thus the orthodox believe! Anathema to him who does not thus believe! Peter has spoken thus through Leo! ...This is the true faith!" - Acts of the Council, session 2
An example of an infallible teaching (a teaching without error) is the doctrine of the Holy Trinity (the triune nature of God: the Father, the Son, and the Holy Spirit). This doctrine was declared infallible at the Council of Nicea (circa 325AD). Now, this is not to say that the Holy Trinity only existed from that point, no. What's this tells us is that at the Council of Nicea in 325AD, those in attendance at the council came to the conclusion that it was absolutely beyond doubt and beyond mere faith that God existed in three natures. Belief in the Trinity was held by the early Christians and the Early Church Fathers (St. Irenaeus, St. Hippolytus of Rome, and St. Origen for instance) and this was a teaching that was commonly agreed upon, but only declared infallible in that year because the council was finally convinced (epiphanised, if you will) that his teaching was truly inspired and guided by the Holy Spirit. A "Eureka!" ("I've found it!") moment if you will.
Other infallible teachings include (in no particular order) the Hypostatic Union (Christ as a fully human and fully divine being; true man and true God), Mary's Immaculate Conception, the Beatific Vision (the eternal and direct visual perception of God), just to name a few.
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So in summary "infallibility" refers to the Catholic Church's teaching authority (the Magisterium) and the Church's capacity to teach infallibly, that is to guarantee that a particular teaching/doctrine is free from error. Papal Infallibility does not mean that the pope himself is infallible or that he is impeccable, no, but under the guidance of the Holy Spirit and through the Church the pope may declare a teaching infallible when he speaks "ex cathedra".
For some further reading and more information on infallibility, have a read of the following article over at Catholic.com: http://www.catholic.com/library/Papal_Infallibility.asp