Is the Pope really infallible?

In recent controversy the pope apologized for insults to Muslims, the concept of jihad, and the prophet Muhammad in a quote. Although I think the apologies were needed questions rise about the pope’s infallibility, if the pope is never wrong then how could he make a mistake? Well, the truth of the situation is that the pope was not deemed truly infallible until 1870, not by God, but by the First Vatican Council but only if he was speaking ex cathedra and is addressing the morals, beliefs, and the church itself. Catholic historian Michael Walsh says that the pope has to make a dogmatic statement in ex cathedra while sitting in the chair of St. Peter for a statement to be infallible, while many pope also (I believe) pat their head and rub their belly at the same time, but that’s just to show of.
The Catholic Church works like a federation, bishops control their diocese, and the bishops answer to the pope who is the bishop of Rome. When a bishop has a moral question or bishops argue over church decisions the as the pope for a judgment who is “infallible” and is prevent from straying from the faith by the Holy Spirit. There’s five years of Catholic school and british news for you, believe me, I know.
*information from http://news.bbc.co.uk/2/hi/uk_news/magazine/5355758.stm