The theorem states that if $U$ is a simply connected open subset of the plane, then there is a biholomorphism between $U$ and the unit disk. I imagine, due to the wild generality of the result, no explicit construction can be expected in general. However, in many concrete cases, I would think a construction "by hand" should be possible; and in applications (to problems in engineering, for example), this would almost be a requirement.
Do you know of such a construction, or of a reference where these constructions are discussed? (The answer may of course only apply to certain families of open sets.)
I know of a very nice reference: "Schwarz-Christoffel Mapping", by Tobin A. Driscoll and Lloyd N. Trefethen, Cambridge Monographs on Applied and Computational Mathematics (No. 8). The Schwarz-Christoffel Mappings explicitly give us biholomorphisms between the upper half plane and the interior of simple polygons. I am hoping for additional examples.