The convolution of two continuous-time signals $x(t)$ and $h(t)$ is

$$ (x*h)(t)=\int_{-\infty}^{\infty}x(\tau)h(t-\tau),d\tau.$$

For each output time $t$, one function is shifted relative to the other, corresponding values are multiplied, and the products are integrated over the shift variable $\tau$.

Convolution is commutative and associative when the integrals 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 signals into a third while preserving their shift structure. That algebraic structure is why it later appears naturally in linear time-invariant systems and Fourier transforms.