It looks like it might be a black marble and those are etch marks from acidic things. Dark marble is known to show etch marks especially badly because etch marks are generally light colored. You could have it professionally re-honed. I am not sure about the basin but if it is brass it is likely a "living finish? meaning that is supposed to age like that. If you are truly unhappy with it, a new vanity top with sink of that size should not be too expensive. You could even find a decent one on craigslist...a lot of time fabricators have them left over or are an order someone changed their minds on.

Unless brass is lacquered it is always going to start oxidizing pretty fast. If you left it it would probably turn a mottled brown black eventually with some green verdigris. It would have to be relacquered to maintain a brighter brass finish.

Thanks so much for the information. Originally I thought this was marble, then I was thinking granite. Either way, it's awful. I will probably end up replacing the whole thing eventually but I was hoping for something to make it look a bit nicer in the meantime.

I used Brasso on the basin and the fumes about did me in. Plus it looked worse than when I started so I gave up on trying to polish it.

Thanks for the help:)

Tracey