Quarter-round is a thin strip of molding that attaches to a baseboard. It is sometimes called shoe molding, since one of its primary purposes is to protect the baseboard from blemishes that can occur from shoe scuffs, vacuums, cleaning equipment and other things that could damage the baseboard. It also functions as a way to hold in place a floating floor, and it covers the gap between floor and baseboard that is caused by minor undulations in the the subfloor. There are two types: shoe molding, which is more oval in appearance, and traditional quarter-round, which is one-fourth of a circle.
Begin in a corner. Cut a 45 degree angle at the end of the quarter-round. Measure the length needed and cut another miter cut at the opposite end.
Nail the cut piece in place to the baseboard, not to the floor. Use finish nails that are long enough to pierce through the molding, baseboard and Sheetrock to the wall studs -- about 2 1/2 inches in length. If the molding is being used to anchor a floating floor, apply downward pressure to the strip of trim as you nail.
Work around the room, using miter cuts at each corner or outside edge.
Use a nail set and a hammer to countersink the nails. Fill the holes with wood putty, wiping away excess with a soft cloth.
Paint the trim piece to match either the baseboard or the floor. It should not look like a separate piece.