Warning: Wikipedia contains spoilers.
Brother Francis Gerard of Utah, a humble novice, is sent from the Abbey of the Albertian Order of St. Leibowitz on a mission of "penance, solitude and silence" to the desert, where he discovers an ancient fallout shelter containing "relics." These relics are handwritten notes on crumbling memo pads bearing cryptic texts like "pound pastrami, can kraut, six bagels -- bring home for Emma"; Brother Francis soon realizes that these notes appear to have been written by Leibowitz himself.
The Albertian Order preserves the knowledge from before the Flame Deluge and the subsequent Age of Simplification. This is a parallel to the role of medieval monks preserving the Classic culture through the Dark Ages.
Miller worked for many years on a sequel, Saint Leibowitz and the Wild Horse Woman; it was completed by Terry Bisson, and published after Miller's death.
The Babylon 5 episode Deconstruction of Falling Stars contains a scene that is almost identical to the premise of this novel.