Most of the best ones have been mentioned. Personal favorites include The Stand (get the expanded, most recent version), Canticle for Liebowitz, Alas Babylon, On the Beach, and The Road. I can only somewhat recommend Lucifer's Hammer at this point because it contains a ton of dated references for a current reader.
Others to consider:
Earth Abides - worth reading because its very realistic and the story covers both the immediate post-apoc but also basically the rest of the main characters life.
Dark Tower series - not fully post-apoc but enough to keep you interested if that is your thing, though not all the books in the series are equally deep in that aspect of the world.
Warday - a mid-80s "travelogue" of a couple of guys traveling around the US after a "limited" nuclear exchange between the US and Soviets. Very much focused on the real effects of even a limited nuclear apocalypse.