Chihuahuas come in two different coat types which define the "varieties". The AKC Breed Standard says that other than coat, the two varieties must be the same. Long Coated Chihuahuas are defined by their long silky coats. They have beautiful plumed tails that are carried high, often curving over the top of the body and by long ear furnishings and long body hair.
Here is a lovely example of a long coated chihuahua.
The smooth coated chihuahua is characterized by its smooth hard coat. All colors and patterns are accepted in both varieties.Here is an example of a smooth coated chihuahua