Mathematical transformations are applied to signals to obtain further information from that signal that is not readily available in the raw signal. There are a number of transformations that can be applied, among which the Fourier transforms are probably by far the most popular. The Fourier transform gives the frequency components (spectral) exists in the signal. But, when the time-localization of spectral components is needed, a transform giving the TIME-FREQUENCY representation of signal is needed. This is in short, if we take the Fourier transform over the whole time axis, we cannot tell at what instant a particular frequency rises. The wavelet transform is a transform which gives this sort of information.
The Continuous Wavelet Transform is a correlation between a wavelet at different scales and the signal with the scale (or the frequency) being used as a measure of similarity. Wavelet transforms enable us to represent signals with a high degree of sparsity. This is the principle behind a non-linear wavelet based signal estimation technique known as wavelet denoising. Wavelet denoising attempts to remove the noise present in the signal while preserving the signal characteristics, regardless of its frequency content.