The book is divided into three parts -- each six centuries after the last -- that show Father, Son, and then Spirit. Much of the book is about science vs. religion. Miller converted to Catholicism and he was strongly in favor of the religion side.
The book starts six centuries post-apocalyptic with a novice named Francis in an abbey of the Order of Leibowitz. It's clear to us that Leibowitz was part of the military/scientific group that provided weapons for the apocalypse. To the monks, he was someone who became a monk and tried to save paper and was hanged for his beliefs. The abbey has kept close guard over pieces of paper for many years when all papers were to be burned and they are also looking for proof that Leibowitz was a saint. Francis is led to find proof of Leibowitz's sainthood by a man clearly meant to be the Wandering Jew.
The second part starts at the capital of Texarkana and is strongly political -- New Rome (in the US) against Texarkana against nomadic tribes. A brilliant scholar comes to the abbey in case some of Leibowitz's (and other scientist's) equations actually have value. He is angry to find out that he is not the inventor, but the rediscoverer of some science. The monks greet him with a rudimentary generator and arc light -- devised by one of the monks -- and he is more gracious to them because of that. As he gets ready to go home, he finds out that the ruler of Texarkana is coming to take the abbey and all the small governments.
The third part takes place at the abbey in a time where apocalypse is imminent and is strongly religious. Many Catholic beliefs are pushed by the current abbot. The abbey receives fallout victims and initiates the secret plan. It's no accident that many of the monks have been spacemen and it turns out that the church has its own spaceship. The men, some scientists, women, and children, plus enough Catholic officials to assure continuation of the faith, and copies of the papers will go to make a new colony and never come back. The last spaceman on board taps Earth dust off his shoes before he closes the door.
There are many many religious clues and symbols throughout the book and if you don't read Latin, you might want this study guide which includes Latin translations.
I had the hardest part with the third section, of course, because choices made by the last abbot cause pain to innocent people. I suppose if you believe in souls, maybe the pain is worth it so you get to heaven, but it's always seemed like torture to me.
There is a sequel, Saint Leibowitz and the Wild Horse Woman, that was mostly written by Miller and finished by Terry Bisson, but it's not very good.