What does that gold drop back mean when Obama gives a speech?

Answer 1

You may be referring to the gold drapes which are often used as a backdrop when presidents give speeches from the Oval Office and from the East Room of the White House. Drapes of this type have been there for more than a decade. There are some internet myths that claim the drapes have "Arabic writing" on them. That is false. The gold ones in the Oval office seem to be solid colored, but the ones in the East Room do have some designs; however, the designs are not writing at all-- rather, they are stylized decorations that are supposed to be doves and olive branches.

As for why there are gold drapes in the background, mostly because that is the color that was chosen when the rooms were redecorated. There does not appear to be any particular reason for that color (as opposed to red or green or some other color); and in fact, over the years, there have been other colors of drapes in various rooms of the White House. Sometimes, First Ladies redecorate, and the decor is whatever they chose for that room.