This series of three articles from DDJ are old, but still very interesting. The Fast Wavelet Transform (FWT) introduces this O(N) transformation which improves "old" FFT O(N logN). A Wavelet Analyzer introduces the recursive computation of wavelet coefficients tree. Here one of the most impressive property of wavelets is expressed in all its power. Each level of the tree is a more detailed approximation of the original data. The Wavelet Packet Transform is an interesting evolution of FWT. If you want to get more information, Wikipedia has more information on wavelets.
In short, wavelet transform enables analysis of data at multiple levels of resolution. This is useful in many contexts. For instance, it has been succesufully apploed to Similarity detection and clustering of images for fast ranking and clustering of similar images.