Assess the damage.
Light scratches require a different repair approach from deep scratches.
For light scratches, lightly rub fine steel wool over the scratch.
Be sure to rub the steel wool along the grain of the wood or you’ll have more scratches to repair.
For deeper scratches, use lightweight sandpaper on the scratch.
Be sure to sand along the grain of the wood.
Rub mineral spirits over the sanded area.
Rub mineral spirits over the area to smooth it out and to pick up the fine dust from the sanding.
Fill the scratch with premixed wood filler.
Use a plastic putty knife to fill the scratch with wood filler that matches the color of the floor. Let the filler dry thoroughly.
Be sure to use a plastic putty knife to apply the filler (you don’t want to create any new scratches).
Sand the filler and wipe up all the dust.
Once the filler has dried, use lightweight sandpaper (use 180 grit). Be sure to only sand the excess filler.
Seal the sanded area.
Refinish the board with varnish, polyurethane, or a similar product to match the rest of your floor.
Warning: Don’t use a foam roller or brush to apply the polyurethane — you’ll end up with air bubbles in the finish. Go with a lambswool applicator or natural bristle brush, instead.