Prime Obsession: John Derbyshire
I can say with confidence that this is the most interesting book for laymen about the Riemann Hypothesis ever written. Derbyshire has written a book that is roughly equal parts Men of Mathematics and introductory number theory. Since he's pitching at a general audience, the introduction to the complex plane and calculus can drag a bit if you've encountered them before, but eventually he's sketching results that you're not likely to have encountered unless you've taken analysis or number theory courses at an undergraduate level. The result is an extremely approachable introduction to the most famous outstanding problem in mathematics.
One area where I felt the book could have done better was in the annotations. As an example, on page 259, Derbyshire refers to two theorems that allow the number of zeta function zeros in the critical strip between two heights to be compared to the number of zeros on the critical line. This is a key component in verifying that there are no violations of the Hypothesis up to a given height above the real number line. It wouldn't be appropriate to his main text to give a detailed explanation, but even in the footnote for the section, he refers to the theorems only as "Theorem A" and "Theorem B". A Mathworld link or some other more detailed reference would have been ideal here.
At 350+ pages, Prime Obsession will keep you tied up for a while. I'd recommend it if you recognize at least 3 of the following 4 names: Euler, Ramanujan, Russell, Hilbert. You will really enjoy the book if you know what an Erdős number is, but don't have one. You will certainly be bored by it if you have an Erdős number less than 4.