The intention behind my question is to understand what are the kind of general problems of which the ideas of Riemann's dissertation (1851) lie at the heart of it's solution methods. In his dissertation, Riemann introduced his mapping theorem (and gave a flawed proof of it based on Dirichlet's principle) and "Riemann surfaces". According to the historical introduction to Carathéodory's "Conformal representation":
In 1822 Gauss (1777-1855) stated and completely solved the general problem of finding all conformal transformations which transform a sufficiently small neighbourhood of a point on an arbitrary analytic surface into a plane area. This work of Gauss appeared to give the whole inquiry its final solution; actually it left unanswered the much more difficult question whether and in what way a given finite portion of the surface can be represented on a portion of the plane. This was first pointed out by Riemann (1826-1866), whose Dissertation (1851) marks a turning-point in the history of the problem which has been decisive for its whole later development; Riemann not only introduced all the ideas which have been at the basis of all subsequent investigation of the problem of conformal representation, but also showed that the problem itself is of fundamental importance for the theory of functions.
What i'm trying to understand is the meaning of "conformally representing a finite (not infinitisimal) portion of a surface on a portion of the plane". As far as i understand, a conformal mapping can be visualized as acting (everywhere on the surface) like a composition of rotation (so angles are preserved) and rescaling (magnification of linear element). It should be mentioned that in general, the rotation angle might vary from place to place, as well as the scaling factor. So, conformal mappings already act on finite portions of a surfaces. So what is the meaning of all the references i find to much more difficult problems?