What I am looking for is a geometrical, or rather intuitive, explanation of why a complex integral of $1/z$ along the unit circle centred at the origin is different from that of $1/z^2$. I understand technical stuff regarding the integral, meaning that I can prove
$$\frac{1}{2 \pi i}\int \frac{dz}{z^m}=\begin{cases} 1 & m=1 \\ 0 & m\ne1 \end{cases}$$