You are going to need a heavy-duty needle, yarn, stuffing, and an old sweater. Except for the sweater (which you can find at a thrift store), you can find all of these materials at a craft store. A good choice of material for the sweater is 100% wool, for warmth and comfort, but any material will do. If you have a sewing machine, you may want to use it, but it is not required. Begin by flipping your sweater inside out. Then, sew along the neckline. (This closes the top of the sweater off so that you can stuff it later.) Finally, turn your sweater out the right way. Visualize a line that moves from one armpit of your sweater to the other. You are going to sew both sides of your sweater together, across this line, using either your heavy-duty needle and yarn (or a sewing machine). Ultimately, you are creating a "tube" that runs all the way from one wrist-opening to the other. After you have sewn the line from armpit to armpit, take your stuffing and insert it into one wrist of the sweater. Keep adding stuffing through the wrist-opening until you have a soft tube, that runs all the way from wrist to wrist. Now touch the wrists together to form a soft circle. This will form the "edge" of your cat bed. Once again, the "tube" of stuffing you have created is going to form the soft, plush edge of your cat bed. In order for this work, you need to sew the sleeves together, forming a loop. First, tuck one sleeve inside the other. Then, using your heavy-duty needle and yarn, sew around in a circle. The "body" of your sweater will now form the bottom of your cat bed (where your cat will lay). Line up the "body" with the plush loop you have created, with the body fabric on the bottom. Any fabric that extends out farther than your plush loop is excess. Remove this excess fabric with scissors. Then, using your heavy-duty needle and yarn, attach the bottom to the loop.

Summary:
Gather materials. Sew up the neckline of your sweater. Sew a line from armpit to armpit. Fill your "tube" with stuffing. Sew the sleeves together. Attach the bottom.