I got the Adafruit power supply, in the description for the product
The black wire is your every day classic +12VDC with center positive. The white wire is the negative supply, its also center positive which means the center is the same as the black wire's ground. The 'ring' of the white-cable connector is -12VDC. You can draw about 500mA from each side, a total of 1A
How should I hook it up then?
Center of black wire is + 12V
Tie the ring of black wire and ring of white wire - GRND
Center of the white wire -12V
So if I follow your schematic, the +12V and GRND goes to the very top +,- rails and the -12V goes to that rail in the middle (as in your breadboard)?