"The Book" in the title of this node refers to
Erdos' favorite book -- the book compiled by
God which contains the most beautiful proofs of every
theorem in mathematics. The book that every mathematician is constantly trying to get a good look at.
"Why do people care about these 'Book' proofs?"
In the words of Hardy (from A Mathematician's Apology), "There is no permanent place in this world for ugly mathematics." While this may not be the case, important mathematics does tend to be pretty and pretty mathematics tends to be important. It is often useful to revisit old results and try to view them from different angles. Some of the most beautiful proofs that we have of certain results aren't the ones that were given originally but ones that have been perfected over years, decades, and even centuries in some cases. Such proofs often give rise to techniques that apply in more general situations and new techniques lead to new results. Gauss's law of quadratic reciprocity is an excellent example of this, I think. Gauss didn't call it his golden theorem for nothing.
Whether or not a given proof is a Book proof could end up being a matter of heated debate. Try not to hurt anyone over it.
Proofs from The Book, written by Martin Aigner and Gunter M. Ziegler, is a book that supposedly gives us a glimpse at a few of the proofs in the Book. Now I don't think it's worthwhile worrying about whether or not they have provided an accurate portrayal of proofs in the Book. What I do know is that they have done a wonderful job with theirs. I would recommend this book very highly to anyone who is even remotely interested in mathematics as it doesn't really require much background on the part of the reader. The book is divided into five sections: Number Theory, Geometry, Analysis, Combinatorics, and Graph Theory -- so there is something in there for everyone.
It would be nice if people would softlink their favorite proofs on e2 here. :)