I just reformatted my PC with Win7 64bit to try and take advantage of 64-bit python processing. However I'm not sure what to do here: Our office runs ArcGIS 10.1, TOAD and SDE for Oracle. Toad says I need to install a 64-bit client before I can connect to Oracle. The ArcGIS Resource Center says before I can connect to SDE I need to install a 32-bit oracle client. I am confused.
You actually have to install both 32bit and 64bit Oracle clients to get everything working properly.
The article you show in your screenshot is referencing ArcGIS desktop (not the background processing), which is still a 32bit program. You will have to have the 32bit client to connect with ArcGIS Desktop and other 32bit programs. You'll also need the 64bit client to connect with 64bit programs (such as Toad, in your case).