New fur coats are very expensive, but used ones can be found for almost nothing at a thrift store or vintage store. These coats usually contain holes, dirty patches or rips, which are not worth the hassle for many people who would rather buy new fur coats. Their loss is your gain if you're willing to put in the time and a few dollars into repairing the coat.
If the tear or hole is uneven, or the fur is dirty in places, carefully cut out the fur until the hole is even and clean. Sewing fur is fairly forgiving, but if you don't have a clean hole to start with, the patch won't last as long.
Thread a leather needle with a cord thread that matches the color of your fur. You can purchase these items at a drugstore or sewing store. A leather needle will be thicker and wider than a regular needle, but it will also be more difficult to press through the fur material. Knot the thread several times at the bottom to keep the sewing from unraveling.
Press the threaded needle from the bottom of the fur to the top, hiding the knot on the underside of the fur. Work as close to the edge of the hole as possible and make your stitches small without weakening the fur. If you have trouble getting the needle through the fur, pull the needle through with a needle-nose plier, which you can purchase at a hardware store.
As you stitch the hole, gently pull any caught fur out of the stitch before pulling each stitch tight. This way, the stitches will be almost invisible underneath the fur.
After you have sewn the hole entirely shut, knot the thread on the underside of the fur by sliding the needle under the previous stitch, then placing the needle in the loop as you are tightening the next stitch. If you are worried about the stitches coming undone, place a small dab of glue on the knot.