Let $X$ be a compact Riemann surface with boundary. Let us shrink each connected component of the boundary into a point. We get a closed topological surface $Z$ with several marked points (which came from the components of the boundary of $X$) such that the complement $Z\_0$ of these points carries a complex structure.



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Rmk. I think that if such a complex structure exists, it is unique.



Shrinking the boundary of a Riemann surface
No. Complex structure on $X$ "knows" whether an end is a hole or a puncture.
The conformal invariant responsible for this is called the extremal length of the family of closed curves surrounding the hole/puncture.
See Ahlfors, Lectures on quasiconformal mappings.


The simplest case is a sphere with a puncture (conformally equivalent to the plane) vs sphere with a hole (conformally equivalent to a disk). These two things are conformally not equivalent. This is essentially Anton Petrunin's remark. 


