Stephen King's "The Stand: Complete & Uncut" is a post apocalyptic book (no zombies). I think it's as awsome as a fiction book can get. I'd recommend it to anyone, and it's right up the OP's alley.

There's also a 4-part movie series made of it 17yrs ago. It's alright, but movies based on Stephen's books rarely ever do the books justice. It's 1100+ pages long, was the first novel I've read of that size, and I can't say enough of what an awsome read it was. T'real.

Will have to check that out :)

If you manage to get your hands on
http://en.wikipedia.org/wiki/The_Last_Day_of_Creation
(or the original, or some translation), you're in for some very interesting reading. (Although technically, it's pre-apocalyptic, since it involves time travel and takes place several million years BC.)

Also,
http://www.gog.com/en/gamecard/original_war
is freely based on the book, and the game isn't bad either. :-)

Not strictly post-apocalyptic, but "Cat's Cradle" by Kurt Vonnegut Jr is an interesting take on Apocalypse, or rather, events that lead to it.

Another not strictly post-apocalyptic book is "Martian Chronicles" by Ray Bradbury. It includes a rather haunting short story set in a post-apocalyptic America.

Are you referring to "There Will Come Soft Rains," perhaps? The relevant Wikipedia page has a link to the short story in .pdf format here: Ray Bradbury, "There Will Come Soft Rains"
Tends to get reprinted a lot. Good collection, good story.

A decent percentage of the works of Philip K. Dick involve the Earth suffering a nuclear armageddon while the rest of the solar system's colonized worlds prosper. There's a few interesting exceptions.
This PKD story is a modern classic, which is to say that it's already been adapted to film very badly. The Second Variety
You might consider looking up his story "Autofac" as well.

I heartily recommend Philip Reeves' Mortal Engines series - not just post-apo, but steampunk post-apo!

Plus there's Harlan Ellison and his "A boy and his dog" and some other stuff. To get a taste of his style, do read his short story "I have no mouth but I must scream". Also a post-apo of sorts.

Oscar mike golf, Krakadyla, I think I love you. Thanks for the link.
Edit: The link's missing a file. When you encounter a missing image, follow this link and you'll find what you missed.

Oh god.
I'm just going to go curl up and try not to think about that story now.

Actually, this link might be better.

Damnation Alley by Roger Zelazny is a MUST read
http://en.wikipedia.org/wiki/Damnation_Alley
Movie in late 70ties was not bad either...

The "Direct to Cable" mini-series for this one is quite decent as well...
Done in early 2000s ... http://www.imdb.com/title/tt0219224/

Here is a link to a real life post apocalyptic story... :)
http://www.kiddofspeed.com/chernobyl-land-of-the-wolves/author.html

A lot of good books listed here. Saw the movie 'A boy and his dog' but never read the book.

A great post apocalyptic book (didn't see it mentioned yet) is Swan Song by Robert R. McCammon. Not hand in hand Fallout but does have a few similarities - well mainly just the faction (Legion).
Definitely worth reading.

Definitely check out "The Postman", "Damnation Alley" and some Philip K Dick ("The Defenders" and "Dr. Bloodmoney" come to mind; there are others).

I'll add:

Lucifer's Hammer -- Larry Niven & Jerry Pournelle

The Last Ship -- William Brinkley

and possibly:

Jem -- Frederik Pohl (partially set on Earth; nuclear war partway through the book)

Philip K. Dick's "The Defenders:"
http://www.gutenberg.org/files/28767/28767-h/28767-h.htm