Ask the questions, "What is interval analysis and how is it useful?", and you will get different answers to each of these question depending on whom you ask.
The basics of interval arithmetic can be taught to most people. To move beyond the basics, and to see how it can be applied to real world problems is more difficult. I would like to share some of the resources that I, as a beginner, found useful in learning about interval analysis.
Provides an excellent rationale and theoretical underpinnings to interval analysis. Builds on R.E. Moore's original 1966 and now classic monograph Interval Analysis.
The book introduces interval analysis in the framework of numerical analysis. It is well written and the mathematics is clearly presented. Also, it provides a nice introduction to automatic differentiation.
Besides, providing an introduction to the field this book is full of interesting applications of interval analysis. Of particular usefulness are the algorithms given for finding global optima for non-linear functions, and the chapter on parametric bounding.