The discoloration is either caused by rust or hard water stains. If you are seeing the same kind of discoloration on your other plumbing fixtures such as rings in the toilet, water stains in the tub, etc. then you have hard water.
Hard water stains are very tough to remove and won't come off with bathroom cleaner. There are products on the market which are specifically made to dissolve the built up minerals such as lime, calcium, sulfur, etc.
If you are seeing rust in your fixture, than that would indicate that it is near the end of its life. The internal components are probably worn too, which will eventually lead to a failure (leak). If I was in that situation, I would just replace the entire fixture. You may be able to buy a new aerator, and a rebuild kit from your local hardware store.
In either case you shouldn't have to replace the whole sink unless it is in bad shape.