The convolution of two discrete-time sequences $x[n]$ and $h[n]$ is

$$ (x*h)[n]=\sum_{k=-\infty}^{\infty}x[k]h[n-k].$$

For each output index $n$, one sequence is shifted relative to the other, corresponding values are multiplied, and the products are summed.

Convolution is commutative and associative when the sums are well defined:

$$xh=hx,$$

$$x*(h_1h_2)=(xh_1)*h_2.$$

It is also distributive:

$$x*(h_1+h_2)=xh_1+xh_2.$$

Convolution combines two sequences into a third while preserving the shift structure of the inputs. That algebraic structure is why it later appears naturally in linear time-invariant systems and Fourier analysis.