Answer
The purpose of this executable file (program) is to verify that Oracle is installed on the system that the post installation is being run against. Without verifying the existence of ORA.dll on the machine, the Post Installation wizard fails to load and throws an error.
The verifyora.exe executable file is installed with the ArcSDE component and is called by the 'ArcSDE for Oracle Post Installation' executable. Running the verifyora.exe executable manually only initializes the Post Installation wizard.
The default location for verifyora.exe is:
For Oracle 10g -
C:\ArcGIS\ArcSDE\ora10gexe\tools
For Oracle 9i -
C:\ArcGIS\ArcSDE\ora9iexe\tools