This can be created by heat or moisture, or from both heat and moisture. Most commonly from a hot coffee cup or from water left on the surface for a long time.
There are two types of finishes that are used on wood. They are either polyurethane based (water) or oil based. I'm not sure what the type of finish is on my surface, but the softened hardness of the finish, smoother finish, and cloudy nature of the stain leads to the probability that it is water based.
This unfortunate little blurb on the internet recommended the following:
Q: One of my relatives placed a hot cup of tea on our wood dining room table and it left a huge white ring? How do I remove the ring from the table?
A: The white ring is moisture that has been absorbed by the finish. When you do get white rings appearing, this is a sign that the finish is wearing out and losing its ability to repel moisture. The heat from this cup of tea softened the finish and allowed the moisture to get into the top layer of finish. In this case, since it is a new white ring, sometimes you can be successful using a hairdryer on the low/warm setting to evaporate that moisture back out. However, if you have a piece of furniture with a white ring that has been in there for some time, we need a little bit of abrasiveness to remove it. I usually get a pad of 4-0 steel wool - the finest steel wool they make. The steel wool by itself is going to leave scratches, so I first pour onto the white ring a finishing product, usually Minwax® Antique Oil or Minwax® Wipe-On Poly, and then use the 4-0 steel wool on top of the oil to lightly abrade the surface. Take a rag, wipe it off, check it, and gradually you'll see that white ring disappearing. Then take Minwax® Wipe-On Poly or Minwax® Antique Oil and go over the entire surface of the table, because that white ring indicates the finish is worn out. It needs another coat of finish to prevent that from happening again.
This recommendation is pretty bad because the stain has probably penetrated the full thickness of the finish which would require the removal of all the finish and probably a pretty ugly refinish if you can't find the exact stain or finish used. Also, I don't have the patience to sit around with a hair dryer to see if that would work.
Try this method which requires next to no skill and with tools you probably have around your house already! This instructable will work with fresh or even really old stains!