The most likely reason is that the system can't find the "notepad.exe" file (or one of the assemblies it relies upon): it worked fine on my system - as I expected it would - so it's probable that you are trying to run a "custom" notepad file (and your comment that "The notepad.exe is near my exe file." would back that up).
So, if the application is in the same folder as your startup EXE file, then specify the path absolutely:
string p = Application.StartupPath + "\\" + "notepad.exe";
System.Diagnostics.Process.Start(p);
And it should work - assuming your "notepad.exe" doesn't need any other assemblies which aren't also in the same folder.