When you wash clothes at home in the washing machine, water is the solvent used to do the cleaning. Many types of fabric, however, do not handle water very well. For example, wool and water just don't mix. There are also many types of stains that water is not particularly good at removing.
In dry cleaning, a petroleum solvent is used instead of water. It is "dry" only in the sense that it is not wet with water. In the early days of dry cleaning, this solvent was often kerosene or gasoline. Today the industry uses a solvent called perchloroethylene almost universally.
The clothes are washed in this solvent, and then the solvent is recovered in an extractor so it can be reused (and so that it does not evaporate into the air and cause pollution). Once the clothes are cleaned, they are pressed so they look like new.
There is a fair amount of controversy around perchloroethylene and its health effects at the moment. Several of the links below talk about the problems it can create. Some dry cleaners are now using CO2.