I recently did a re installation of Fallout 2 and only installed Killap's Restoration Patch. At first I could not get the game to run without running it as an administrator and once I was able to run it. The game ran in windowed mode but another window— A blank window shows up next to it. The game works fine of course. I don't know if anyone has ever encountered this problem before but any suggestions or help is appreciated.

Try starting the game directly with FALLOUT2.EXE in the game's installation folder (..\Fallout 2\).
If it doesn't run, run it explicitly as Administrator by right-clicking it and select Run as Administrator.

In the game's folder is also a file called f2_res_config.exe to set up graphics.
Run it and set the following:
- Uncheck "Windowed"
- Check "Scaling X2"
- Graphics Mode: Direct Draw 7 (if it doesn't work then DirectX9)
- Screen Resolution: Set this to your Desktop resolution.
- Colours: 32 bit
- Close with "Done"
Start it again and check if the everything is saved. If not, you have to run it as Administrator.

And just in case, don't install the game into Program Files, but always in a custom folder, like C:\GOG Games\

I followed your suggestions for the f2_res_config.exe but it just sets my games to full screen. I actually want the game to run in windowed mode, just without the extra window of course.

Ah, okay, then run it again and check the "Windowed" option.
Is the second Window appearing now?

Try setting compatibility mode for Windows XP to FALLOUT2.EXE
Right-click it, select Properties and go to the Compatibility tab. Select the right mode and check the option to run it as Administrator. Apply the settings and run FALLOUT2.EXE by double-clicking it.

Wonderful! It worked. Thank you very much friend.

You're welcome!