One of the most confusing and misunderstood Catholic teachings amongt non-catholics and catholics alike is the theology of Papal Infallibility.  The easiest way to explain Papal infallibility is to explain what it is not.

 

Papal infallibility does not mean:

 

....that the Pope is all knowing and is impeccable in his personal opinions.  This has never been taught by the Catholic Church and is not what Papal infallibility means.  The Pope is just a human like the rest of us and his knowledge on certain subjects may not be any greater than the average person sitting in church on Sunday.

....the Pope is free from sin.  Every Pope since and including Peter was born a sinner and can commit personal sins like the rest of us. Even the Pope is required by Church Precepts to confess his sins.  Also, in light of this it is true and a fact that the Catholic Church has had Popes in the past who were what some people would call "Bad Popes".  This however in no way distracts from the office of Pope (Head Bishop) which Christ established with Peter.  Even Christ choose men who he knew were sinners (for example - Peter denied Christ and Judas betrayed him) to be his first apostles.

....the Pope is infallible every time he speaks.  The Church teaches that the Pope is not infallible every time he speaks nor are all his personal writings considered infallible.  When the Pope does speak infallibly he is guided by the Holy Spirit (see Matt 10:16-20, Matt. 16:17–19; John 21:15–17) to come to this truth.  The only time the Pope makes an infallible statement is when he speaks for and with the Church (this is called "Ex Cathedra" or "From the Chair") on matters of faith and morals.  It should be mentioned that many Popes throughout history never made an infallible statement.

Here's a couple of statements on Infallibility:

Vatican II explained the doctrine of infallibility as follows: "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. This authority is even more clearly verified when, gathered together in an ecumenical council, they are teachers and judges of faith and morals for the universal Church. Their definitions must then be adhered to with the submission of faith" (Lumen Gentium 25).

The Catechism of the Catholic Church states:

[CCC 889] In order to preserve the Church in the purity of the faith handed on by the apostles, Christ who is the Truth willed to confer on her a share in his own infallibility.  By a "supernatural sense of faith" the People of God,
under the guidance of the Church's living Magisterium, "unfailingly adheres to this faith." [CCC 890] The mission of the Magisterium is linked to the definitive nature of the covenant established by God with his people in Christ. It is this Magisterium's task to preserve God's people from deviations and defections and to guarantee them the objective possibility of professing the true faith without error. Thus, the pastoral duty of the Magisterium is aimed at seeing to it that the People of God abides in the truth that liberates.  To fulfill this service, Christ endowed the Church's shepherds with the charism of infallibility in matters of faith and morals. The exercise of this charism takes several forms:

[CCC 891] "The Roman Pontiff, head of the college of bishops, enjoys this infallibility in virtue of his office, when, as supreme pastor and teacher of all the faithful - who confirms his brethren in the faith - he proclaims by a
definitive act a doctrine pertaining to faith and morals ... The infallibility promised to the Church is also present in the body of bishops when, together with Peter's successor, they exercise the Supreme Magisterium," above all in
an Ecumenical Council.  When the Church through its supreme Magisterium proposes a doctrine "for belief as being divinely revealed," and as the teaching of Christ, the definitions "must be adhered to with the obedience of
faith. This infallibility extends as far as the deposit of Divine Revelation itself.

Some more on the Nature of Infallibility:

The idea that the Holy Spirit can speak infallibly through the Pope is denied by non-catholics.  Non-catholics insist that no man can be infallible when he speaks.  This however contradicts what they believe about the Holy Bible.  All non-catholics will say claim that the Bible is the infallible Word of God and that it contains no errors.  When they say this they overlook the fact that it was men who wrote the Bible.  So, if the Holy Spirit used men in an infallible way to write the Bible then is it not out of the question that the Holy Spirit could speak infallibly through his Church via it's visible head - the Pope?  If non-catholics claim that no man could ever speak or write
infallibly then they must also believe that the Bible is not infallible since men wrote it!

Bible Verses that elude to the theology of infallibility:

“And after there had been much debate, Peter rose and said to them, "Brethren, you know that in the early days God made choice among you, that by my mouth the Gentiles should hear the word of the gospel and
believe.” (Acts 15:7)

"And I tell you, you are Peter, and on this rock I will build my Church, and the powers of death shall not prevail against it. I will give you the keys of the kingdom of heaven, and whatever you bind on earth shall be bound in heaven, and whatever you loose on earth shall be loosed in heaven’" (Matt. 16:18-19).  Teaching a non-truth would mean that the powers of death (hell) have prevailed against the Church.  

"And Jesus came and said to them, "All authority in heaven and on earth has been given to me. Go therefore and make disciples of all nations, baptizing them in the name of the Father and of the Son and of the Holy Spirit, teaching them to observe all that I have commanded you; and lo, I am with you always, to the close of the age.”  (Matt 28:18-20) The Church which Christ founded will always be around and it cannot teach errors!  This Church must be here today unless Jesus lied, which is not possible!  

A question for Non-Catholics to consider................... As mentioned above all non-catholics will agree that the Bible is the infallible word of God and that it contains no errors.  At the same time they will claim that the Catholic
Church is not infallible in it's teachings.  When they do this they are forgetting that it was the Catholic Church who put the books of the Bible together.  So one question these non-catholics should ask themselves is this:
How can something which they believe to be subject to error (the Catholic Church) produce the Bible which is free from error?  Remember that something fallible cannot produce something infallible!  So, if the Catholic Church could teach error then that would mean that the Bible could also contain errors since it was canonized by the Catholic Church!

Something else to ponder: Non-catholics do not believe in the doctrine of infallibility and they would argue that no man can act in an infallible way.  These people seem to forget that the Bible was written infallibly by the
Biblical authors.  If these men were not infallible in their writings then the Bible cannot be considered to be free from error and thus inspired.  The fact is that the Biblical authors were guided by the Holy Spirit just like the Pope
is when he declares a teaching to be infallible.  If the Holy Spirit could guide the Biblical authors infallibly then why couldn't He continue to guide his Church through the charism of infallibly?  Also, since Saint Peter was the first Pope non-catholics must admit that he acted infallibly when he wrote his epistles which would mean that at least one Pope acted infallibly!