The classic book on inequalities is Inequalities by Hardy, Littlewood, and Pólya, first published in 1934 (ISBN 0521358809). I’ve mentioned it a few times before. The book is quite thorough. I imagine more people use it as a reference than read it cover to cover.
The Cauchy-Schwarz Master Class by Michael Steele (ISBN 052154677X) is a more recent book on inequalities, published in 2004. The preface explains what the title means by a “master class.”
In the fine arts, a master class is a small class where students and coaches work together to support a high level of technical and creative excellence. This book tries to capture the spirit of a master class while providing coaching for readers who want to refine their skills as solvers of problems, especially those problems dealing with mathematical inequalities.
The Cauchy-Schwarz Master Class book lives up to its stated aim. Reading the book and working through the exercises reminds me of taking music lessons. The emphasis is on learning techniques rather than cataloging specific results.
The final book I’ll mention is When Less is More: Visualizing Basic Inequalities by Claudi Alsina and Roger B. Nelsen (ISBN 0883853426). Like The Cauchy-Schwarz Master Class, this book emphasizes problem solving technique. More specifically, it emphasizes geometric techniques for understanding and proving inequalities. (I’ve written a review of When Less is More, but unfortunately you have to log in as an MAA member to read it.) When Less is More is as concerned with beauty as with truth.
You can tell from the cover art and the architectural allusion in the title that this book cares about aesthetics. Each proof is a polished gem.