Ok, on windows you can use GDI, DirectX and OpenGL to create games.

GDI should not be used for any games in my opinion, it is not hardware accelerated and on vista due to the software / hardware bliting it has become even slower! If you try to make a game in this you will quickly hit the wall. GDI is meant to be used for creating windows applications, not games.

DirectX is only availiable on Windows and XBox, DirectX isn't in my opinion as good as OpenGL however OpenGL has a horrible plugin system. OpenGL can do more but DirectX can do more nativly. The plugins are really inconveniant requiring differant versions for each graphics card manufacturer. OpenGl can be used on Linux, Mac, Wii, PS3 in addition to windows. So thats another advantage.

OpenGL is implemented by differant vendors, therefore some are open source and some are closed source. The capabilities of the card are of coarse only known when the program runs. GDI is completly seperate, however if you want so some reason you can hybrid DX and GDI but the hardware/software swap is slow.