Wavelets literaly little waves are mathematical signals with good inner properties that are used to decompose and synthesize digital signals with a smooth and lacunary representation. Imagined in the years 1980s by J. Morlet, their theoretical background was set and popularized after the works from Y. Meyer et S. Mallat with the aim to fill the gaps opened by Fourier Analysis. Wavelets are hidden in the background of numerous innovations that most people use in their everyday life such as the jpeg2000 image compression scheme, the Mpeg4 video codec or image processing algorithms in tomography.
Mathematical background
From a math point of view, a wavelet is a function with following properties
 oscillating
 smooth (continuous, derivable) under the Hölder meaning
 localized both in time and in frequency
This latter property is thoroughly discussed in the section about the timefrequency plane hereafter.
The Meyer wavelet
A wavelet instance with infinite smoothness is due to Y. Meyer and is formally defined by its Fourier transform
featuring the intermediate mappy
The Meyer wavelet spanned hereafter in the space of the physical variable and in the Fourier space is clearly oscillating, smooth with an infinite class of differentiability and is well localized both in time and frequency domains.
Other wavelet patterns are illustrated in preambule

Doglet2 or mexican hat, infinite smoothness, well localized but weakly oscillating

Daublet2 or Daubechies wavelet with compact support

Coiflet3, an almost symetric variation on the theme that holds additional trivial moments at the price of a longer support

Spline4 designed by BattleLemarie from spline interpolative polynomials
Timefrequency plane
An other interesting representation is to illustrate a wavelet in the timefrequency plane that stands for the time axis horizontally and the frequency axis vertically. Drawing Meyer wavelet for instance highlights a Heisenberg box which characterizes the occupation area of the Meyer pattern in the timefrequency plane.
As a summary, a wavelet is said to be well localized in the timefrequency plane when its occupation area is as narrow as possible. Heisenberg uncertainty principle shows that there exists a minimum area reached solely in the case of doglet pattern.