There are essentially three types of fog machines. the first is commonly referred to as "rosco" fog and is white cloud like smoke generated that eventually disperses into a haze. this is generated buy a machine that superheats a special fluid, usually glycol based, and shoots out a nozzle.
the second is what is known as a Hazer, this works essentially the same way but it comes out as a haze to begin with. these are normally used in rock and roll so you can see the beams of the moving lights.
the third is a dry ice fogger, which uses water and dry Ice to produce a low lying ground cover type fog. There are also chiller units that will cool Rosco fog in an attempt to make it stick to the ground but in my experience (which is mostly theatrical) they don't work very well. The colder it is in the room the more the fog will stay low.
All three of these can be easily rented from any theatrical supply house. Cheap versions of the the basic fog machines can be purchased from most Haloween costume places or online stores. The fluid can also be purchased online.
I have also heard folks talk about using smokers and smoke candles that are used for finding leaks in AC systems. these can be purchased from places like Grangier and McMaster Carr. but I have no idea how well they work for photography - they strike me as difficult to controll.
For a thick fog that stays up I would thing you would want something like This
and you will need to fire a a controlled burst every few shots
I did a shoot for a band recently where I used a Reelfx DF-50 hazer, which is the workhorse standard in the concert world. We were shooting in a smalish rehearsal space and it didn't take much to fill the space. So you might be able to really let it go for a while and get the look you are going for. The other nice thing about the DF-50 is that it does not leave any residue on your gear. (or anything else). Some of the older oil based hazers and some of the DIY solutions leave a horrible film on everything.Here is a sample shot
haven't quite figured out how to post images here yet.
No matter what system you use the key to getting fog to look right is temperature and air flow. You want the air in the room as still as possible (get some fans to clear it when necessary) and warm for high level fog and cool for low level fog. And don't forget about smoke alarms. You may need to disable them.
Hope this helps.