What the heck. Everyone else is doing it, so I'll be a lemming.
If you are a Gamer, you should read these books:
The Hobbit and The Lord of the Rings by J.R.R. Tolkien. Pretty standard for just about everyone, I'd say. Hugely influential.
The collected Conan stories by Robert E. Howard. Very easy to find these days, which is a nice bonus.
The collected Fafhyrd and the Gray Mouser stories by Fritz Leiber. Actually, this is my favorite swords and sorcery series, even bypassing Conan.
The collected Elric stories by Michael Moorcock. Emo mage-emperor wanders the world.
The first three books of The Song of Ice and Fire series by George R.R. Martin. Not sure I'd bother after that, as I'm 99% sure he's going to die before he finishes the series, and this is a decent stopping point.
The Malazan: Book of the Fallen series by Steven Erikson. I personally love this series, though I have not yet read books 9 and 10.
The Deed of Paksennarrion by Elizabeth Moon. The best portrayal of a paladin in print. Nothing else is even close.
The Vlad Taltos books by Steven Brust. Psychic assassins in an elven empire. If that doesn't sell it, nothing will.
All of the various Newford books by Charles de Lint. He's the best urban fantasy author out there.
Neuromancer by William Gibson, so you can understand cyberpunk before it got twisted into the emo crap it is being pushed as now. This is the seminal book, and it's a straight-up shadowrun without magic.
Downbelow Station by C.J. Cherryh. My favorite science fiction book.
Starship Troopers by Robert Heinlein. Space marines vs. bugs. Need I say more?
There's lots of other great books out there, but I think these are the best of the lot. Many Forgotten Realms, Eberron, Shadowrun, and Warhammer novels are actually quite good, though more than a few are really, really bad.