Measure the width of the mattress for which you want to build a headboard. Mark the width on a piece of plywood, adding or subtracting a few inches depending on if you want the headboard slightly larger or smaller than the bed.
Draw a 5-foot tall square or rectangle on the plywood, using the bed measurement for width. Cut the shape from the plywood using any kind of saw, such as handsaw or jigsaw.
Draw a shape on the back of the veneer that matches the plywood shape. Cut the veneer with a utility knife and place it over the plywood to check that the pieces line up exactly. Use fine sandpaper to even up any uneven areas on the plywood or veneer. Remove the veneer from the plywood and set aside.
Cover the front of the plywood with a thin, even layer of wood glue. Place the veneer over the glue, lining up the corners precisely. Use a hard rubber-backed roller and roll over the veneer to remove air bubbles and distribute the glue evenly underneath the veneer. Press the veneer firmly against the plywood and insert a staple. Place the staples 1/2 inch in from the edge, positioning them horizontal to the edge and spaced every 5 inches. Staple all four sides of the veneer to the plywood board. Let glue dry.
Measure out enough matching 1-inch wide veneer molding to cover the top and sides of the headboard. Peel the paper from the back of the molding and carefully press it along the edges of the headboard to cover the staple marks. Let molding dry.
Cover the front and sides of the headboard with a thin layer of clear varnish. Let dry. Flip the headboard over and cover the back with a thin layer of the varnish to help prevent the headboard from scratching the wall. Let dry.
Mount headboard to the bottom of your mattress frame with screws or use hooks on the wall to hang the headboard as if it were a giant picture frame.