There have been many good suggestions here, so forgive me if I am repeating a few.
Here are some books that I would highly recommend in no particular order.
FlatLand: Creates a real understanding of dimensions. Great book for challenging your thinking. Also be sure to pick out the movie. This could truly be 2001 for math geeks.
A Mathematician reads a newspaper: Goes through a newspaper, and explains the math behind it. All topics covered, politics, business, lifestyle, and much more.
Innumeracy: What happens if you don't know math? Total societal collapse. Okay maybe that's a little extreme, but this subject is important, and everybody should understand it.
Why do buses come in threes": A personal favorite. Shows how math plays into everyday life. Touches every subject. This book is interesting, informative, and amusing. I highly recommend it.
How long is a piece of string: Sequel to the previous book. Not quite as good, but still better than most.
Conned Again, Watson: Where else can you have Sherlock Holmes explaining probability and statistics to a poor unlearned Dr. Watson.
A History of PI: This is more of a history book than a math book. But it is a history of math, or specifically pi.
The Joy of pi: Like the previous book, but less serious.
Euclid's Window: Now it is really time to bend the old mind. History, Adventure, and non-Euclidean Geometry. Great stuff.
Hyperspace: You thought 4 dimensions were bad? How about 10.
A Mathematician's Apology: Since I started with a classic. I will also end with a classic. G. H. Hardy's book is a must-read for any serious math aspirant.