A cigarette stain on a bathroom sink is actually not a stain but damage caused by heat. As a result, removing it is more of a repair than a question of scrubbing. You will have several options depending on the type of sink you have. It is important that you determine what your sink is made of before you start working on the stain.
Try bleaching out the stain. If you are not positive that the stain is from a cigarette, first treat it as a mineral stain. These are the easiest to remove and can be lifted by covering the area with a rag saturated with bleach for about an hour, then scrubbing. It is definitely worth a try before you break out the big guns.
Try an abrasive cleaner. Wet the area, then scrub it with a cleaner like Ajax. Use firm, circular motions and rinse frequently so you can see if you are making progress. If the results are good, then keep scrubbing until the stain is gone. It could take awhile. Watch for damage to your sink, though, since abrasive cleaners can be hard on the finish and the seal.
Try sanding off the stain. If the stain has blistered, you will need to sand the entire area. This will harm the finish on your sink, so be ready to do some damage control. Start with the lowest-grit sandpaper, which is also the roughest. Once you have sanded off the brown, damaged area, use progressively finer grits to polish the area until it once again gleams like glass. You will have removed any finish or seal that was on the surface of the sink, so you will need to address this issue.
Reseal the damaged area. Use the paintbrush to apply a small amount of sealant to the area that you repaired. You will need to use a sealant that is compatible with your sink material, so check the label before you start sealing. Apply the sealant evenly and allow plenty of time for it to dry--at least 5 hours--before you use the sink.