What to do?  

It's what I call "bloom".
Usually caused by moisture and/or heat.
Try testing the finish to see if you can tell what it is. Put a single drop of denatured alcohol or your preferred whisky on a less conspicuous area, let it sit for 10-20 seconds, then blot it away. Immediately test the spot for stickiness or any sign that the alcohol has begun to dissolve the finish. If it has, it's likely shellac and the bloom can easily be removed by a quick French-polishing of the top. Any mobile furniture repair service person should be able to do it in about 15 minutes.
If it's not shellac, the process is a bit trickier, but sometimes not impossible to repair without refinishing. Liberon makes a product called "Burnishing Cream" that's meant to work on this type of damage. I haven't tried it, but their products that I have used are of reliable quality.
http://www.restoration-materials.co.uk/www.restoration-materials.gbr.cc/...

.
"White marks indicate that water has penetrated the surface of the finish. An easy method of removing white marks is to use mayonnaise. Cover and rub the white mark with mayonnaise and leave overnight and it will magically disappear overnight."
from:
http://www.essortment.com/furniture-repair-removing-stains-marks-househo...

I just bought
a table with some white rings and durn it I'm trying the mayonnaise trick! It sounds plausible and I am all about plausible!

I have used a paste
of cigarette ash and some rubbing and it worked very well. I have also been told the same with baking soda as well as the mayo. The other I have heard is heat. Have not tried that yet.

I can't explain it,
but I'm suddenly hankering for a grilled chicken sandwich with lettuce and burnishing cream on whole wheat toast.

"I can't explain it,
but I'm suddenly hankering for a grilled chicken sandwich with lettuce and burnishing cream on whole wheat toast."
me too now you mention it.

I'll try the mayo first
I've heard about that trick before, but am sceptical. I also heard about using heat to remove the marks - i'm a little hesitant about that one!
Thanks for your advice, tktoo. I thought you'd like this:
http://www.videojug.com/film/how-to-make-steak-sandwich-with-garlic-mayo

Ooh, Daniel, is there a DA food network?
Don't make me post my grilled portabella marinade recipe. You'll be sorry.
BTW, the day you find me slathering condiments on fine furniture is the day you can nail me into a pine box.

Ever seen Leather CPR?
I have a sneaking suspicion that it's really just mayonnaise.

I didn't have you down as a pine man, tktoo
Are you also not curious as to whether the Mayo trick actually works? I am.
I'll be back tomorrow, asking how to remove oil marks.

mayo
didn't work for me. Think there is a finish prevent absorption. Guess I will try the hot iron and cloth that TCHP indicated which I THINK i have done successfully previously.

I bought this.
It was next to the Burnishing cream.
Reckon it'll taste lovely with my poached salmon.
http://www.liberon.co.uk/retouch-and-revive/ring-remover,434,464.html?&a...

Don't forget...
To be generous with the dill.

Admittedly, I'm more of a rosewood and teak type
tempered by a Naugahyde and pine income, alas.
Please pass the Grey Poupon!
Edited to change "gray" to "grey". I nearly forgot that this is an English-speaking crowd.
http://www.youtube.com/watch?v=ho61-oaU6wo&feature=endscreen