Wavelet analysis is a technique to transform an array of N numbers from their actual numerical values to an array of N wavelet coefficients.
Each wavelet coefficient represents the closeness of the fit (or correlation) between the wavelet function at a particular size and a particular location within the data array. By varying the size of the wavelet function (usually in powers-of-two) and shifting the wavelet so it covers the entire array, you can build up a picture of the overall match between the wavelet function and your data array.
Since the wavelet functions are compact (hence the term wave-let), the wavelet coefficients only measure the variations around a small region of the data array. This property makes wavelet analysis very useful for signal or image processing; the "localized" nature of the wavelet transform allows you to easily pick out features in your data such as spikes (for example, noise or discontinuities), discrete objects (in, for example, astronomical images or satellite photos), edges of objects, etc.
The localization also implies that a wavelet coefficient at one location is not affected by the coefficients at another location in the data. This makes it possible to remove "noise" of all different scales from a signal, simply by discarding the lowest wavelet coefficients.
For a general introduction to the wavelet transform and its applications see Hubbard (1998).
The IDL Wavelet Toolkit uses the continuous and discrete wavelet transforms. Details on the discrete wavelet transform can be found in Daubechies (1992) and Mallat (1989). A good introduction to the DWT and multiresolution analysis is given in Lindsay et al. (1996).
The DWT routines are based on the routines described in section 13.10 of Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge University Press), and are used by permission.
An introduction to the continuous wavelet transform for time series analysis can be found in Torrence and Compo (1998), along with a discussion of statistical significance testing.