As opposed to the functions sine and cosine used for Fourier transforms, a wavelet not only has locality (small support) in the frequency domain but also in the time or spatial domain. Therefore, wavelets look like fading in and fading out waves (hence the name).
The simplest wavelet is the Haar wavelet. A wavelet commonly used in the natural sciences is the Morlet wavelet. There is also the Daubechies wavelet which is well-suited for data with fractal properties.