Tukukkk

I remember a book where it took place during a nuclear war, and these people destroyed books and other things and killed people who knew how to read. Then it skipped 600 years forward three times and introduced three central characters. All these characters were Catholics. What is this book called and who is it by?

This is likely A Canticle for Leibowitz by Walter M. Miller Jr.

After 20th century civilization was destroyed by a global nuclear war, known as the "Flame Deluge", there was a violent backlash against the culture of advanced knowledge and technology that had led to the development of nuclear weapons. During this backlash, called the "Simplification", anyone of learning, and eventually anyone who could even read, was likely to be killed by rampaging mobs, who proudly took on the name of "Simpletons". Illiteracy became almost universal, and books were destroyed en masse.

Isaac Edward Leibowitz had been a Jewish electrical engineer working for the United States military. Surviving the war, he became dedicated to preserving knowledge by hiding books, smuggling them to safety (booklegging), memorizing, and copying them. Eventually, whether by Leibowitz' intention or simply through the sanctification of repetition, his supporters and followers became a monastic order, the "Albertian Order of Leibowitz", dedicated to preserving knowledge. The Order's abbey is located in the American southwestern desert, near the military base where Leibowitz worked before the war, on an old road that may have been "a portion of the shortest route from the Great Salt Lake to Old El Paso". Leibowitz was eventually betrayed and martyred. Later beatified by the Roman Catholic Church, he became a candidate for sainthood.

Six hundred years after his death, the abbey still preserves the "Memorabilia", the collected writings that have survived the Flame Deluge and the Simplification, in the hope that they will help future generations reclaim forgotten science.

The story is structured in three parts: "Fiat Homo", "Fiat Lux", and "Fiat Voluntas Tua". The parts are separated by periods of six centuries each.

For a roughly analogous (but completely different and desperately weird) short story, you might look up Stephen Baxter's Inherit the Earth in Traces. This isn't the droid you're looking for; but if you're into post-apocalyptic Catholics (of a sort), then this might scratch your itch.