I often see here formula expressed in term of $z$. But what is $z$?
$z$ is interpreted as the time advance operator. $z^{-1}$ is the time delay operator.

So for a difference equation like
$$ y[n]=a x[n]+b x[n-1] $$
in the $z$ domain
$$ Y(z)=(a+ b z^{-1}) X(z)$$