Sinks get a lot of abuse over time. Dishes are pile up in them, silverware is throwing in carelessly and metal pans bang against the sides. No matter what type of material your sink is made of, this constant wear and tear will eventually cause some scratches. You may believe you can't repair the scratches if you have a stone material for your sink, such as a granite composite. However, in most cases, the damage is repairable with a little bit of effort.
Lay a No. 000 steelwool pad on the counter over the scratch. Rub slightly in a circle pattern for a few second and then pick it up. If the scratch is still visible, repeat the rubbing until it is gone. If you have a deep scratch, start with a No. 00 pad instead.
Rub a second time with a No. 000 pad if you started with the coarse No. 00. Rub in a circle as before, working the scratch area and about one-half inch further out into the sink. This will blend the area with the surrounding sink.
Rub with a Np. 0000 pad to buff out the circular pattern made by the other pads. Rub the surface in straight lines, moving the pad in different directions as you go. This will smooth out the circle pattern in the sink. Buff out with a cloth when you are done.
Apply an epoxy glue to chips and place them where they came from in the sink. Hold it down with your finger for a few seconds until the glue sets up somewhat. Allow to dry for several hours.
Smooth excess glue off of the surface with a No. 0000 pad after the epoxy has dried.
Scrub the entire sink with progressively finer wool if there are stains or many scratches. This will smooth the entire surface and sand away stains.
Wipe on an acrylic finish for granite. Apply it to the buffed area or the entire sink if you sanded down the entire surface. Use a lint-free cloth, and allow it to dry for a couple of hours. Apply a second coat once it is dry and third if the container suggests it. Let the sink stand without use for a day to ensure that all the coats have set up.
If you have a crack in the sink, it is best that you call a professional. Water damage from leaks could cause even more damage, and the entire sink may need replacing.