If I create a model in 3D Studio MAX with animation and want to use it with DirectX, then how it is possible?
you can convert it into an x file and then use the d3dx mesh loading functions to use the x file. You can use teh tools that come with the directx sdk to export either a maya file to an x file or a 3d max file to an x file.
What about the .X Files? To how many formats it can be exported to be used?
You can export an x file to any format really...as long as you know how an x file is arranged and how the file your exporting it to is arranged.
How do professional programmers do this?
Most games have their own proprietary mesh format. The programmers would make a mesh in maya of 3d max and then convert it into their own formats. For example if I were using teh quake engine, then I could make my stuff in maya and convert it to an md2 (or was it md3?) file.
There are many many ways to go about it.