I studied the classification theorem for compact and connected surfaces a few years ago. The proof that I was told and that is presented in many books uses representations of quotients of polygons. I sketch it in the following paragraph.
Essentially, you begin with an arbitrary polygon with some quotients between its sides (which represents a compact and connected surface) and you show that the representation is homeomorphic to another one of a sum of tori and projective planes (or homeomorphic to a sphere). Finally, you show that a sum of tori and projective planes is homeomorphic to another one if both are orientable (or both are not orientable) and they have the same Euler characteristic.
I am interested in the two first lines of the previous paragraph, that is, "If I am given an arbitrary connected, compact surface why can I assume that it is homeomorphic to a polygon with some quotients between some sides?". Another formulation of the same question is " Why has a compact, connected surface a representation as a quotient in a polygon?". It seems to me that it is the property of being compact what ensures that you can do this, but I can not see clearly why.
Thank you very much.
What you are looking for is the triangulation theorem for surfaces. See here for a discussion.