These are any region where you have a strong difference in intensity as you move in any direction from one pixel to another. If you keep the extremely blurred version of the smooth areas along with a few of the edges, you can get a beautiful rendering of your original image. Compression is only the first half of the procedure. After transmission, text, images, and data must undergo reconstruction, otherwise called decompression. Then, you keep multiplying the pixels by four, and hopefully put back the edges.
Reconstruction is completely equivalent to compression, but much faster.
You can get progressive renderings, with a handful of coefficients. Wavelet compression has yet to make a major impact on the transmission of images. But researchers have no doubt that its time will come, probably very soon. International committees that develop standards for Internet traffic have decided to include wavelet compression in the new JPEG-2000 standard, which will soon be finalized. Research is also under way on extending the concept to moving images.