The idea of wavelets is to introduce useful bases for the Hilbert space. Useful means useful for harmonic analysis or PDE.
The traditional method for solving such problems is to use the Fourier transform. While this is extremely powerful in many situations, it does have some disadvantages, mostly due to its nonlocality. That is, a particular function f may have localized support, but it is not easy to see this in terms of its Fourier transform f. Wavelets deal effectively with this issue.
There are many applications of wavelets both in theory (they simplify the proofs of many problems) and practise (e.g. compressing information, such as jpg files).