There are cert ainly more characteristics that mathematicians use for defining which wavelets are the most useful, but these two characteristics are what make wavelets in general a very exciting area to explore. The study of wavelets generally takes a strong mathematical background, but a simple Haar-based lossless compressor is a good place to start for those who haven't yet taken Calculus XVII and Super Advanced Linear Algebra. Wavelets are used for many reasons.
Wavelets allow signals to be decomposed into various parts, some of which can simply be thrown away.
With wavelet-based compression, such blockiness is minimized, thus allowing more of the image to be thrown away without suffering consequences in terms of image quality. However, wavelets have another key characteristic. They have the ability to store an image at many resolutions, while minimizing overhead. Next, instead of transmitting the entire version of a more detailed image, only the information necessary to derive a higher resolution image from the low resolution image is translated. As more of the image is transmitted, the decoded image increases in quality.