All of these transforms are trying to take a given signal and find ways to construct it using "atoms" taken from a "dictionary". By looking at which atoms of t

All of these transforms are trying to take a given signal and find ways to construct it using "atoms" taken from a "dictionary". By looking at which atoms of the dictionary you used, you get an idea of what the signal is made up of. The discrete cosine transform is awfully similar to the discrete fourier transform. They are conceptually the same and just have minor technical differences. The DCT and DFT's dictionaries are made up of sine and cosine wave atoms at different frequencies. These waves are all infinitely long, so they work best at representing signals which are repetitive over their whole length. They don't do a very good job at noticing where interesting parts of a signal are. A wavelet is like a small chunk of sine/cosine wave. Instead of being infinitely long, it only exists over a small length of time/space. The wavelet dictionary contains wavelets in different places as well as at different frequencies. So when you decompose using wavelets they do a better job of encoding where the interesting bits are. Wavelets assume your signal might be periodic in particular areas, where the DFT tries to find periodicities over the whole signal.