There are three common ways to produce the fog that you often see in stage productions and at dance clubs:
Fog machines and fog juice are the most common. The basic mechanism is simple -- the fog juice is heated to create smoke. When you overheat oil on the stove and create a lot of smoke, you are doing approximately the same thing. Cooking oil has a tendency to get gummy and smell bad, however. Fog machines use glycerin or glycol mixed with water.
According to a Seattle Times article, "The fog is created using propylene glycol and triethylene glycol, mixed with 20 percent water." One thing that is not clear is whether this fog has any side effects on people's lungs. It seems like it is a problem for asthmatics, but nothing has been proven conclusively for the general population.
If you would rather be safe than sorry, you can use dry ice (which is available at most ice manufacturing plants and is fairly inexpensive -- look up "ice" in the phone book, and never handle the stuff with bare hands!). When you place dry ice in hot water, it creates a dense fog that clings to the floor. This fog contains carbon dioxide and water vapor, so it is completely harmless.