I recently read Poincaré's Prize: The Hundred-Year Quest to Solve One of Math's Greatest Puzzles by George C. Szpiro. I recommend it highly. Some time back I recommended another book on the same topic, The Poincaré Conjecture: In Search of the Shape of the Universe by Donal O'Shea. If you can only read one book on the topic, I recommend the Szpiro book.
Both authors are fine writers. The books are of similar length. O'Shea's book is 200 pages followed by 72 pages of supplementary material: endnotes, two glossaries, a timeline, and an 11-page bibliography. Szpiro's book is 262 pages followed by 32 pages of endnotes and bibliography. Each book provides a different interesting aspect of Poincarés life: Szpiro's book relates Poincarés career as a mining engineer, in the course of which he exhibited great personal courage and deductive ability worthy of Sherlock Holmes to investigate a mining disaster. O'Shea spends a fairly lengthy chapter on the Klein-Poincaré correspondence which has been put forth as an example of the way academics can cooperate even when their countries are mortal enemies. O'Shea's careful reading shows the antagonism simmering beneath the surface of their "polite" academic discussion.
Szpiro introduces a great deal of the mathematics that led to the proof of the conjecture by Grigori Perelman in 2002, almost 100 years after Poincaré made the conjecture. He illustrates the math by very clever analogies, avoiding any attempt to go to deeply into the mathematics, which it seems to me is the only way to present material of such awesome complexity and abstraction to a lay audience (in which group I include myself.)
Like O'Shea, Szpiro shows mathematicians warts and all, as he discusses priority disputes such as the Smale-Stallings-Zeeman controversy of the proof of the Poincaré conjecture in higher dimensions (which preceded the proof of the original three-dimensional conjecure). He is not afraid of picking sides: He argues that Smale deserves credit for the proof, but that his abrasive personality made it difficult for him to get help in establishing priority.
Nor is Szpiro shy in assigning full credit for the final proof to Perelman, though standing on the shoulders of many giants, especially William Thurston and Richard Hamilton. Perelman has been pictured as an eccentric loner, refusing the Fields Medal and the $1,000,000 Millennium prize for no good reason. Szpiro sees him as a man of utmost integrity and great friendliness to those who share his seriousness. It is not surprising, then, that Szpiro takes the great Chinese mathematician Yau Shing-Tung to task for pushing the claims of his students Cao and Zhu, who wrote a paper in which they claimed to given the first real proof of the conjecture, based on Perelman's "outline".
If you are interested in mathematics, you owe it to yourself to read either Szpiro's or O'Shea's book on the Poincaré conjecture.