There are a lot of resources available on the subject of failure analysis. Even if we confine ourselves to failure analysis of structural components using the body of knowledge acquired by mechanical and materials engineers, there is an overwhelming amount of published content.
The classic work Understanding How Components Fail by Don Wulpi was recently revised by my longtime colleague on ASMI’s Failure Analysis Committee, Brett Miller. This book does an excellent job of helping the reader learn to identify and classify different types of damage in metallic components. The emphasis is on mechanical damage. The book does not require much metallurgical background. It's great for those new to the field.
For people who do failure analysis on a regular basis, I'd suggest a monumental effort, in which I participated, that culminated in the 2002 version of the Metals Handbook, Volume 11 on Failure Analysis and Prevention. This book was edited by Roch Shipley and Bill (William T.) Becker. The approach that we took in writing the 2002 edition of Volume 11 differs substantially from the one it replaced. Both are worth having. The older version is valuable for its treatment of categories of components (shafts, fasteners, gears, etc.) and industries (chemical manufacturing, petroleum refining, automotive, etc.) The new book presents, in a fresh way, the mature thought of many people who had been working in the field for decades.
For people who have to understand corrosion, Mars Fontana’s classic Corrosion Engineering is the book to get. Fontana was head of the Metallurgy Department at The Ohio State University. I think he deserves credit for creating a system of classification of corrosion types that is essentially still the foundation of the field today. While understanding all the major types of corrosion in all the alloys used in every industry is essentially impossible, Fontana’s system breaks all corrosion into six or seven basic types. This is adequate for most of my work. When it’s not adequate, I have to do research or contact a colleague with greater expertise in this area.
If you have to review other's failure-analysis work (even occasionally), I'd use the short and readable How to Organize and Run a Failure Analysis by Dan Dennies. This book provides a way to gain the foundations of the organizational aspects of a failure analysis. One of the reasons I like Dan's approach is that he makes it clear that you have to think about your data, and there's no cut-and-dry way to interpret data from a failure analysis. It's inherently different from doing certification test work.
I also really like a newcomer to the field, In Plant Failure Analysis by Neville Sachs. It is geared toward people who don’t necessarily have a metallurgy lab. It helps the general maintenance and operations employee gain an understanding of what information they can extract from a visual examination of a “failed” machinery component. It also gives guidance on when a more detailed investigation would be helpful. This is also a good book to help the failure analyst with a materials background understand more about the lives of various machinery components.