Any engineer or mathematician will be familiar with fourier transforms which in engineering are used to extract frequency information from an amplitude/time signal. The limitation of fourier is that it tells you nothing about when in a signal the high or low frequency components are found. Wavelet transformations provide information resolved in time as well as frequency.
My understanding comes mainly from reading this site from Rowan University’s electronics department.
A sample waveform (mother wavelet ) is multiplied, integrated and energy normalised across the input signal with its position along the input signal time base being varied (time resolution) and with the width (scale) of the mother wavelet being varied (frequency resolution).
The transform is generally reversible so that the original signal can be recovered, and like fourier a digital version is available and actually requires fewer calculations than the fourier.
What I am not clear on from a quick scan of the literature on the web is how you choose a mother wavelet, clearly the function describing your wavelet has be tractable within the maths of the transformation but there are several possibilities. The abstract of Moortel et al suggests that a tighter wavelet (such as the Paul) will give better time resolution and a broader multipeaked mother wavelet like the Morlet will give better frequency resolution. This seems perfectly logical, but I wonder whether in practice what you do is choose a mother wavelet whose shape most closely matches that of the signal you are looking for in the data.