I had that happen once in a rental, it was sooo annoying. Every time I put a cup of coffee down the sink, it was stained. Or tomatoes, or wine, or tea, or anything but water or milk!

I used Iron Out to get the discoloration out (I think, against the mfr recommendation, after much trial and error). I bought a sink mat (there are decorative ones that are quite cute), knowing that the sink was clean, and I couldn't do anything else. The sink mat actually made me feel much better about the ugly sink. They are about $7 and come in all kinds of interesting and colorful designs...pebbles, flowers, lace, tomatoes...

If the finish is gone on your sink, and ONLY if the finish is gone, use ZUD scouring powder. I buy it at hardware stores. Recommended to me by a plumber, but it is so strong that I would never use it on a new sink for fear it would scrub the finish off.