I recommend this book highly. It explains the history, context, and importance of the Poincaré conjecture, as well as many of the attempts to solve the problem, culminating in the successful solution by Grigory Perelman. The writing style is lively, and the explanations seem to be about as clear as possible, given the complexity of the mathematics, the amount of material covered, and the relatively short length of the book (200 pages + 75 pages of footnotes and appendices). The book is semi-popular; while the target audience is probably people with an undergraduate degree in mathematics or the equivalent, there is enough historical and personal narrative to appeal, I would think, to the non-mathematical reader. The book ends with a page-long appreciation of the civilizations and people who erected an edifice of thought that culminated in the statement and proof of this conjecture, which makes a major contribution to the understanding of the universe in which we live.
O’Shea shows how the history Poincaré conjecture is the history of much great mathematics, extending from ancient to modern times, and including such intellectual giants as Euclid, Gauss, Riemann, and Poincaré, and such stellar contemporary mathematicians as Milnor, Smale, Freedman, Donaldson, Thurston, Yau, and Hamilton.
For me, a major side benefit of reading this book was its recommendation of Jeffery Weeks’ book, The Shape of Space, which explains many of the big ideas of modern topology and geometry while demanding little mathematical background, only a willingness to think hard and work at visualization.