Andrew Wiles heard about Fermat's last theorem when he was 10 years old, and decided that he was going to prove it. He finally succeeded 30 years later. Fermat's Enigma by Simon Singh tells the story of this proof. Singh tells of how Fermat claimed to have a proof, and of how in the following centuries many people also thought that they had proved the theorem, only for a fault to be found in their work. The attempts did lead to a lot of interesting mathematics though. Thus Fermat's conjecture became more and more famous, and several prizes were offered for its proof.
Although quite a bit of the book deals with the history of mathematics, it isn't one of those books where the author tries to relate as much material as possible to one topic - Singh's book seemed to keep pretty much to the point. Thus there is a chapter on the Taniyama-Shimura conjecture, and quite a bit of the book describes Wiles and how he set about proving the theorem, as well as how he corrected it when a fault was found. The book doesn't really get into the mathematics. There is a bit of maths in the appendices, but even that is fairly simple. Singh has thus written a very readable book which can be enjoyed by a wide readership.