I am wiring a window A/C unit but not sure how to wire the plug. I am running 12-3 to a 20amp double breaker. The plug is a 20 amp 250 volt that has one plug up and down the other is side to side with the ground on the bottom. (Looks like someonw winking). There are only two screws on the side in which to attach the wires. Do the black and red go on one together and if so, which side?

Black and red to the two gold screws. It doesn't matter which.
Ground to green.
Cap off the white, but tie it in at the panel.

You could have saved yourself some money and used 12/2 and used the white and black as your two hots. You just have to re-mark the white with a permanent marker.
This is the most common way to wire a straight 240v circuit.

You only need "3-wire" if the circuit requires a neutral. This would be a 120/240v circuit.

I agree with Stan!