To help clarify a well known characterization: If U is a connected open bounded simply connected planar set, then the boundary of U is a simple closed curve iff the boundary of U is locally path connected and contains no cut points.
0) Definition. p is a cut point of the connected space X iff X\p is not connected.
1) Definition. By `boundary' of an open bounded planar set U, we mean the difference U closure minus U.
2) For necessity, observe that the unit circle X is locally path connected and X\p is connected for all p in X.
3) For sufficiency, every orientation preserving conformal homeomorphism ( Riemann map) f: int(D) --->U extends continuously to the respective closures iff the boundary of U is locally path connected. (Caratheodory). If the extension is not 1-1, a straightforward geometric construction yields a non cutpoint on the boundary of U.
4) Mirko's nice sketch of the `Warsaw circle' shows a non locally path connected boundary can have no cut points, and pac-man with a closed mouth shows a locally path connected boundary can have cut points. Thus two the mentioned conditions are independent.