Convolution is a mathematical operation which generates a third signal from two input signals. In practical audio terms, you can think of one input as the signal applied to a loudspeaker while the second input is the response of the loudspeaker to a single-sample impulse (the Impulse Response). In that case, (except for the hopefully minor effect of non-linear distortions created by the loudspeaker, such as harmonic and intermodulation distortion) the output of the convolution will be the actual response of the loudspeaker to the applied input signal.

Convolution is an invertible function, that is, from the output and the impulse response the input signal may be reconstructed perfectly. Similarly, from the output and the input signal, the impulse response can be reconstructed.

A more detailed (but still accessible) definition and illustrations can be found here.