I happen to work in the furniture industry. In fact, I manage the finishing operations for a high end furniture manufacturer in NC. So I'll offer you some free expert advice.
The fact that the damage was done by nail polish remover (whiuch is acetone) pretty much narrows down the list of possible finishes on that piece. It's most likely a standard nitrocellulose lacquer, or a pre-catalyzed nitrocellulose lacquer. Odds are very high that it's the former. NC lacquer is the most common commercial furniture finish in use today and has been the standard for generations.
In the furniture industry, acetone is often reffered to as "wash-off" because it is used to wash the finish off a piece if you screw up the finishing process. Good news is there is a way we can determine whether it's pre-cat lacquer or standard lacquer and both can be fairly easilly repaired if you know what your doing. Here is how to proceed:
-Get an aerosal can of lacquer based sanding sealer from the hardware store, get a can of regular lacquer as well. Preferable semi gloss based on the pictures.
-Get some 320 grit sand paper and a sanding block. The block should be a fairly hard material and perfectly flat on one side. Wood or very hard rubber would make a good block.
-Sand the defect fairly well with the 320 keeping the block flat on the surface. Try to sand the defect and the area a few inchest around it. Sand it well, but not so much that you cut through the finish, in particular the underlying color coats. Your not trying to remove the defect all at once.
-Spray the sanded area with a pretty heavy coat of the sealer. Let it dry.
At this point the sealer will lay down smooth and stick. If it peal, crackles, or otherwise shows signs of poor adhesion, your table has pre-cat lacquer on it. If that's the case, go by a couple gallons of acetone and wash the whole thing off. Feel free to get it soaking wet with acetone and rub the finish off with a rag. It won't hurt a thing. When your done, very lightly sand the piece with 320 and refinish.
If the sealer seemed to stick, proceed as follows:
-sand the repaired area again with the block and 320. sand aggressively in order to cut down the sealer you just sprayed, but again not so deep you remove color..
-Keep repeating this spray and sand process on the repaired area. What you are doing is gradually filling up the pit the nail polish made and then sanding it smooth. Almost like using body filler on a car.
-When the area is filled you'll see it while sanding as the paper will cut the surface evenly smooth.
-Now you can take the lacquer you bought and recoat the whole top for a nice even finish.