If $A \in PSL_2(\mathbb{R}$ then $tr^2(A)>4$ means its hyperbolic. There is a unique geodesic in the hyperbolic plane that is invariant under $A$ and $A$ acts a translation along that geodesic. If $tr^2(A)<4$ then $A$ is elliptic. It has a unique fixed point in the hyperbolic plane and it acts as rotation about that fixed point. If $tr^2(A)=4$ then it could be the identity, or it could be parabolic. If it is parabolic it has a unique fixed point on the circle at infinity in the hyperbolic plane. If you put that fixed point at infinity, then it looks like translation $z\rightarrow z+c$ and with scaling you can make that $z\rightarrow z+1$.

If you take a quotient of the hyperbolic plane by the group generated by $z+1$ you get an open once punctured disk. You should think of this as an open neighborhood of the point
"missing" from the surface.

If you have a surface of finite type, that is it is homeomorphic to the result of removing n points from a closed oriented surface, then the space of all conformal structures on that space is larger than the space of all structures with parabolic ends which is larger than the space of structures on the closed surface.

Here is an elementary example. The conformal maps from $S^2$ to itself are exactly the group of Mobius transformations. You know a Mobius transformation is determined by where it sends three points. Let $C$ be the space of choices of 4 points from $S^2$. Two choices are equivalent if and only if there is a Mobius transformation taking one to the other. I can choose to send three points to three points, but I have no control of where the $4$th point goes so the space of conformal equivalence classes is one complex, or two real dimensional.