I can tell you of at least three applications related to audio.
Auto-correlation can be used over a changing block (a collection of) many audio samples to find the pitch. Very useful for musical and speech related applications.
Cross-correlation is used all the time in hearing research as a model for what the left and ear and the right ear use to figure out a sound's location in space (this is called sound source localization). In the case of two microphones you would cross-correlate the left channel with the right channel.
Convolution is used in simulating reverberation. A room's impulse response can be determined from measurements and that impulse response can be convolved with any sound source to simulate the reverberant response (at the impulse response recording's exact location).
I know this answer isn't complete but maybe it can give you some idea that there is in fact a practical use for auto- and cross- correlation!
So in general, auto-correlation can be used to extract properties of a signal, cross-correlation can exploit the information between two related signals, and convolution can be used to modify the properties of an incoming signal based on some time, frequency, and phase response specified by the impulse response you're convolving the source with.