Title: What is mathematics?
Author: Herbert Robbins and Richard Courant
This book is a very nice introduction to mathematics, it covers basic number theory, analysis, algebra, geometry and topology.
I'am very surprised that i couldn't find it on this list already.
(from a duplicate answer - feel free to edit) This would be for someone who has some mathematical ability, and really wants to understand what math is. Courant goes through essentially all of mathematics, starting at a very elementary level, but getting to some very deep and important stuff. He often does real proofs, and doesn't dumb it down, but does explain things conceptually very well, including sometimes giving just ideas or justifications for really difficult things, like the prime number theorem. I use this when I teach our senior proof seminar, just to force the math majors to own a copy.