The integration of the product of two functions over a range of time offsets. If an input signal is convolved with the impulse response of the signal, the resulting signal is the same as if the signal had passed through a physical system with the same impulse response (or frequency response). Convolution in the time domain is equivalent to multiplication in the frequency domain, which is equivalent to addition in the cepstrum domain. (Source - http://zone.ni.com/devzone/nidzgloss.nsf/glossary)
In terms of sonic transformation, convolution has many potential musical applications, including forms of spectral and rhythmic hybriding, reverberation and echo, spatial simulation and positioning, excitation/resonance modelling, attack and time smearing.