"Dry cleaning" is a type of cleaning process for clothes like suits and sweater. You don't do this at home; you take your clothes to a "dry cleaner", which is the name of the business.
The phrase "dry cleaning" can mean the process of cleaning clothes:
Conventional dry cleaning uses a lot of nasty chemicals that are harmful to the environment.
It's also what you call the clothes that have been cleaned. For example:
Can you pick up my dry cleaning on your way home?