OpenGL is a standardized graphics interface for use between programs and your graphics card. A proprietary version of this by Microsoft is known as DirectX. LWJGL and JOGL are two different Java bindings to OpenGL libraries, both of which have native code they load in.
LWJGL is much more aimed towards gaming with OpenGL, where as JOGL is more worried about complete and perfect bindings to OpenGL/OpenCL. I would recommend LWJGL if you're just getting started.
You're going to have to rewrite your rendering code, yes. But assuming you designed your project well, you shouldn't need to rewrite the game logic. JFrames will probably also have to go, but I'm not completely sure about that.