Steve Jenkins says cheese can be classified into the following varieties:

Fresh: Uncooked and unripened cheeses such as Italian Mascarpone, cream cheese, and French fromage blanc made right after the milk has turned to curd.

Fresh Ripened: The easiest cheeses to make, they are fresh, white, and have no rinds. Examples are Chèvre, Boursin, and Ricotta Salata.

Soft-ripened: These "bloomy rind" cheeses are inoculated at the curd stage. Examples include Brie and Camembert.

Washed Rind: The outside rind of these "smelly cheeses" has been wiped with a liquid such as wine, beer, or brandy. They can be up to 3 months old and Steve thinks these are the great cheeses of the world. Examples are Pont le Vec, Liverot, and Gorgonzola.

Semi-soft: Chaume and Fontina are examples of this variety.

Semi-firm: Appenzella and Swiss cheese are included in this variety.

Hard: These cheeses are generally aged for 2 1/2 to 4 years and are pressed with weights during the aging process to extract the whey. Common examples include Parmigiano-Reggiano and aged Cheddars.

Note that goat cheese and sheep cheese are not categories unto themselves!