DirectX is an API (Application Programing Interface) used for developing video games and other such media applications, much like OpenGL.
The differences between DX9 and DX10 is that Anti-Aliasing is supported in DX10 and DX10 supports shader v4.0 Anti-Aliasing is used to smooth the edges of objects in a video game and makes little difference in games playing in low resolutions. With it on, it will noticeably slow down performance in certain games while delivering an almost unnoticeable difference in visuals.
The difference between DX10 and DX11 is that DX11 supports Tessellation. Hypothetically, DX11 is supposed to enhance performance with compatible systems as well as providing a graphics increase with a feature called tessellation.
DX11 is used with Windows7 and 8.1, DX12 is Windows 10 only.
An example of the graphical improvements with tessellation on can be viewed here: http://www.youtube.com/watch?v=KtbUWiZ48a0 (discontinued), tessellation is turned off at first. Too much tessellation can cause terrible lag just like too much anti-aliasing.
This video has nothing to do with flight simulation, but shows the differences with and without tessellation.