Dry ice fog is still used occasionally for theatrical productions, but is becoming rather rare. It is expensive and a hassle. It does have some desirable characteristics, but usually just isn't worth it.
[Only part of the fog from dry ice is actually carbon dioxide gas. The visible fog is from ambient water vapor that is condensed by the extreme cold of dry ice. Without plenty of ambient water vapor, the fog output of dry ice is reduced. This is another reason why dry ice foggers often involve hot water.]
The idea is simple - drop a chunk of dry ice in some water; it will bubble and smoke. In practice, this is a little more difficult, because the dry ice will cool down the surrounding water and start to form a shell around itself of water-ice. Eventually the smoking and bubbling stops. The only way to keep this from happening is to keep the water hot, which increases the rate at which the dry ice is consumed. That's why dry ice isn't really used as often as people think it is.
Theatrical fog machines for dry ice usually take the form of large metal drums filled with water. An electric heater keeps the water warm. The dry ice is placed in some kind of basket, over the water. When you want fog, you lower the dry ice into the water. It rapidly melts, spewing out fog.
Also, please see the section on dry ice warnings; this stuff can be dangerous.
Another nice feature of dry ice is that it is portable - it does not have an extension cord, or need a bulky machine to make it smoke. {At least you don't need a machine to get some smoke out of it. You do need equipment, often with an extension cord, to keep it smoking over a long time.}
The portable nature of dry ice fog makes it popular for making a smoking punch bowl.
Information on this machine is available at:
http://www.lemaitrefx.com/Fog%20Individual%20pgs%20Price%20List/Page%208.PDF
The machine itself lists for $685. But that's not a factor, because you only need to buy it once. It consumes 20 pounds of dry ice, to provide 5-10 minutes of operation. The price of dry ice varies wildly by location. But figure $1/lb. So best case would be around $2/minute of operation. And you better use it soon, because dry ice evaporates in the machine whether you are making fog or not. And I doubt that this machine has remote electric operation. See that handle on the side? I think the deal is, you want fog, you walk up to the machine and twist the handle, which lowers the dry ice into the hot water.
In order to reduce the possibility of injury, do not handle dry ice with your bare hands, or allow to come into contact with any exposed skin. This is especially true for damp skin, because the dry ice will use the moisture to freeze itself to you.
This is a shame, because it sure would be nice to just combine water, dry ice, and a glow stick to get portable glowing fog!
In practice, this is a little more difficult, because the dry ice will cool down the surrounding punch and start to form a shell around itself of water-ice. Eventually the smoking and bubbling stops. You might get away with fishing out the little ice-balls and dropping in fresh dry ice every 5 or 10 minutes.
All of this takes a back-seat to the safety issues:
Dry ice in your drink looks really impressive.
Just don't do anything stupid and hurt yourself!
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