What you want is something that is accoustically transparent. In other words can you blow air through it easily. Mostly folks use one of four things:
1. Burlap...cheap kind of rough looking, can be bought raw or dyed.
2. Felt...cheap and soft looking, have a wide variety of solid colors available.
3. Lose weave cotton fabric. Wide price range and can be found in all sorts of patterns and solids.
4. Guilford of Maine Accoustic Fabric. Quite expensive but with a classy corporate interior kind of look.
For items 1 through 3 to play it safe you should buy some spray-on flameproofing. Guilford of Maine fabrics are already treated.
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...and on the installation issue if you are using a frame then treat it like a picture (duh). If you are not using a frame use impaling clips. You can also DIY some impaling clips by pounding long nails through a 1 inch board, ideally at a slight upward (when installed) slant. Then screw this board into the wall so the nails are pointing out...impale your unframed covered panel on this.