A brace is what you attach to the back of flats to hold them up.
They're usually made of a triangle of 2" by 3" CLS timber.
You attach them to flats by screwing a block of timber into the frame of the flat and then putting a screw (3 by 10 is best) through the brace. This should be be done at the top and bottom.
You can attach hinges to braces so that the brace will fold away. This allows to to either store the flat more quickly or, more commonly, to angle the brace so it's less visible to the audience.
Always ensure that the bottom of the brace and the bottom of the flat are aligned such that the flat stands at 90 degrees to the ground. If you don't the flat will lean and look kind of weird.
If you're doing a whole line of flats then it's best to screw them all together first and then attach braces. This will avoid the previous problem.
Once the braces are attached and the flats up use stage weights to stop it moving.