We still have those in one of the older bars in my city (Calgary), and they were around in my elementary school's washrooms as well. It's a towel indeed, and it dispenses in a rotating roll. You pull down on it to get the clean portion, much as you would a regular paper towel dispenser, and the used part goes up the backside.

Look up "cloth towel dispenser" or "continuous cloth towel" or "cloth roll towel". Here's a simple diagram of how it works: http://www.target-ltd.com/images/darmdiagr.jpg

Surprisingly few sources on this classic method of hand drying, but some companies still exist, such as the following:

http://www.darmanco.com/endura2.asp

http://mspconverting.com/brands/natures_linen_industrial.htm

http://www.alsco.com.au/services/industrial/continuous-towel/