What is a cusp neighborhood corresponding to a parabolic Möbius transformation in a Riemann surface?

Question

I am referring to this wikipedia entry.

So what I understand is that they are defining it using the Fuchsian model. If $\Gamma$ is a Fuchsian group, its parabolic elements correspond to the cusps of $\mathbb{H} / \Gamma$. I do not know why, or what a cusp mathematically is, but I have seen pictures.

Then there is the statement in the WP entry about $\gamma(U) \cap U = \emptyset$. This does not make sense. What if $\gamma$ were a (rational) rotation around $i$? Wouldn't $\gamma(U) \cap U \neq \emptyset$ in that case?