In order to fix the fur on a stuffed animal you would need to do some preparation. First take a good look at the damaged area. Has the fur fallen of from the fabric or is it torn and the stuffing is showing?
If you have the second issue, usually a simple restitching would be enough. If the torn bit is on a seam, use a thread in the same color for restitching and run along the seam.
If the torn part is on a piece where there is no seam, it is best to use a transparent colored thread or one in a color that is as close to the color of the fabric as possible. Try to do a neat seam as possible.
If the fur has simply fallen off, you have a balding stuffed animal. You will need to buy some fake fur for replacement. Now comes the time for a decision. If you have a balding stuffie, chances are it is quite old and you would find it difficult to find the exact same fake fur for replacement.
But if it was damaged during play for example and the toy is newer, it is possible. Either way you would have to decide whether to go with fur that it is close to the original as possible or change the damaged spot with completely different fur. This would give your stuffie a slightly altered look but ultimately you have the final say.
Ok, now it is time to fix the fur on the stuffed animal.
You have your piece of fake fur ready. One possibility is to trace the seams of the entire damaged piece and rip them to remove the piece. Then use the piece for a template to cut the new fur into the needed shape. Then simply sew the new fur in place with a proper colored thread.
If the old fabric is thin or you don’t want to remove it, you can sew the new fur onto the balding spot. This could work for smaller fixes but you would have to be careful not to make the fix too thick or to look out of place. This is why it is usually best to cut the damaged piece and replace it completely.