I heard the phrase "Dry Camping" while watching an American RV show. What does it mean? Are there similar but different terms for the same thing in other parts of the world?
This term is pretty vague. This paragraph describes it pretty well however:
Dry camping means different things to many people. The common definition I hear from people is camping in a location where electric/water/sewer connections are not available at each site. This definition is very broad and includes everything from camping at a commercial campground with centralized facilities to driving out into the woods and pulling off into the trees.
You may have more luck looking up dispersed camping.
Dispersed camping is the term used for camping anywhere in the National Forest OUTSIDE of a designated campground. Dispersed camping means no services; such as trash removal, and little or no facilities; such as tables and fire pits, are provided. Some popular dispersed camping areas may have toilets.
In the UK, the form of camping you are referring to is known as Wild Camping.
This is pitching up where there are no electrical, water or sanitary facilities. The term applies equally to camping in tents and parking a motorhome / camper van.
Note though that in England and Wales, the practice is illegal. Except in some of the National Parks. However, it is legal in Scotland, except for certain protected areas, like the banks of Loch Lomond.
Just for reference: in thru-hiking terminology, dry camping is camping far from any water source. It basically means that you have to carry all the water you'll need for the evening and the next day (until you meet water).