Wavelet analysis is similar to Fourier analysis in that it can be viewed as a technique for reducing a complicated time varying function to a (hopefully) small number of functional components. Although Fourier analysis can be seen as a special case of wavelet analysis, the advantage of general wavelet analysis is that the basis functions are localized in time, which is a common property of components of time varying signals. There are many applications for wavelet analysis, but we mainly regard them as a useful tool for understanding time series and a means of generating useful features for machine learning models.
Recommended Books

Wavelet Methods for Time Series Analysis
Donald B. Percival and Andrew T. Walden
 Intext exercises
 Answers to some of the exercises
 Errata etc.
We like this book because it has a detailed mathematical exposition of wavelet analysis, it's geared toward statistics, it has a lot of neat applications, it's mostly selfcontained, and it has solutions to some of the exercises. We dislike this book because most of the exercises are theoretical, and there is no real help with coding applications. We recommend looking through the figures available at the above website to get a sense of whether this book is appropriate for your application, as it is a relatively challenging text.