Measure the length and height of the space where you plan to hang the backdrop. Use a measuring tape to find the exact length and height.
Go to a fabric store and purchase a piece of fabric that is slightly larger than your measurements. You may want to buy two pieces of fabric in case you make a mistake the first time around. Also, be sure to choose a good color. For photographers especially, some colors might wash out the subject.
Cut the fabric based on your measurements. Be sure to leave a little extra length and height so that you can adjust if need be.
If you are using the backdrop for photography, purchase a rod that spans the length of the space. Hang the rod by inserting screws into the ends of the rod and affixing to the wall.
Place the fabric on the rod. You may need to sew the top part of the fabric together to create a sleeve for which to pass the rod through. If you are using the backdrop for theater, hang the fabric over the curtain string or affix by either folding over and sewing or by using safety pins.
Test the backdrop by placing a subject in font of it and taking a few pictures to gauge the result. For theatrical purposes, shine stage lights against the backdrop to see if there is a good balance of color and light.