For Catholics, the doctrine of Christ's divinity is central, for it is like a skeleton key that opens all the other doctrines. Catholics have not independently reasoned out and tested each of the teachings of Christ received via the Bible and the Church, but believe them all on his authority. For if Christ is divine, He can be trusted to be infallible in everything He said, even hard things like exalting suffering and poverty, forbidding divorce, giving his Church the authority to teach and forgive sins in his name, warning about hell (very often and very seriously), instituting the scandalous sacrament of eating his flesh—we often forget how many "hard sayings" he taught!
When the first Christian apologists began to give a reason for their faith to unbelievers, this doctrine of Christ's divinity naturally came under attack, for it was almost as incredible to Gentiles as it was scandalous to Jews. That a man who was born out of a woman's womb and died on a cross, a man who got tired and hungry and angry and agitated and wept at his friend's tomb, that this man who got dirt under his fingernails should be God was, quite simply, the most astonishing, incredible, crazy-sounding idea that had ever entered the mind of man in all human history.
The argument the early apologists used to defend this apparently indefensible doctrine has become a classic one. C.S. Lewis used it often, e.g. in Mere Christianity, the book that convinced Chuck Colson (and thousands of others). I once spent half a book (Between Heaven and Hell) on this one argument alone. It is the most important argument in Christian apologetics, for once an unbeliever accepts the conclusion of this argument (that Christ is divine), everything else in the Faith follows, not only intellectually (Christ's teachings must all then be true) but also personally (if Christ is God, He is also your total Lord and Savior).
The argument, like all effective arguments, is extremely simple: Christ was either God or a bad man.
Unbelievers almost always say he was a good man, not a bad man; that he was a great moral teacher, a sage, a philosopher, a moralist, and a prophet—not a criminal, not a man who deserved to be crucified. But a good man is the one thing he could not possibly have been according to simple common sense and logic, for he claimed to be God. He said, "Before Abraham was, I Am", thus speaking the word no Jew dares to speak because it is God's own private name, spoken by God himself to Moses at the burning bush (Exodus 3:14). Jesus wanted everyone to believe that he was God. He wanted people to worship him. He claimed to forgive everyone's sins against everyone. (Who can do that but God, the One offended in every sin?)