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Convolution is a commutative, associative, distributive operation between two functions that produces a a third function. It is defined in the continuous domain as

$$(x \ast y)(t)=\int_{-\infty}^\infty{x(\tau)\space{}y(t-\tau)}\space{}d\tau$$

And in the discrete domain as

$$(x \ast y)[n]=\sum_{k=-\infty}^\infty{x[k]\space{}y[n-k]}$$

Its identity is the Dirac delta function $\delta(t)$ in continuous domain, and the Kronecker delta function $\delta[n]$ in discrete domain.

Reference: Convolution.

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