An unfortunate byproduct of having cats is their penchant for scratching. In the case of homes with carpet on the floors, this can result in a number of unsightly scratch and tear areas. Fortunately, there are ways to go about remedying and restoring these areas to their original condition.
Cut out a piece of leftover carpet from when you first had the carpet installed. If you don't have any leftover carpet, find a hidden place, such as a closet or under a bed, and use a utility knife to cut out a small piece of the carpet.
Cut out the scratched area with the utility knife. Remove the damaged piece of carpet.
Vacuum the hole where you cut out the damaged carpet.
Cut a small piece of the replacement carpet out, ensuring its size fits the hole you cut in the carpet.
Glue in the replacement piece of carpet using carpenter's glue. Press the carpet into the hole and hold firmly until the glue sets. If you can see the seam when you are standing, glue in a couple of extra carpet fibres to disguise it.
If the scratched/damaged area is too large for you to fix, contact a carpet professional, as he will have the tools/resources to remedy the situation. However, he might also recommend installing new carpet altogether.