Main categories of cheese

Answer 1

Cheese is classified by its ripening process (how it is aged) or lack thereof. Within these broader categories, cheese can be classified by its origins, its age (and, consequently, its degree of softness or hardness), its milk(s), and whether or not the milk has been pasteurized.

Different cheese making traditions will have slightly different classification systems due to the use of different methods. Here is an attempt at a fairly general system of classification based on ripening process:

Fresh - Cheeses which are not aged, like fresh chèvre, queso fresco, etc.

Pasta Filata - Stretched-curd (mostly Italian) cheeses which develop long, thin curds (think string cheese), like mozzarella, provolone, etc.

Bloomy-rind - Soft-ripened cheeses to which exterior mold is deliberately added. This mold is white, soft, and thick, preventing the escape of moisture, and causing the cheeses' interior paste to break down over time, becoming softer rather than harder with age. Brie and camembert are two well-known examples.

Brainy-rind - Cheeses which develop surface mold naturally. They are often small, and lose moisture relatively quickly, going from cloudy and soft to quite hard. Crottins de chèvre (those little goat cheeses that look like brains, hence the name) are a famous example. The term "natural rind" can also describe these cheeses, and can describe any other cheese whose rind is not added or specifically encouraged by the cheese maker, which is the case for many tomme style cheeses, which I've included in the "Pressed/Firm" category.

Washed rind - Soft-ripened cheeses which are regularly washed in some liquid, often a flavor-imparting liquid (brine, wine, beer, marc), to encourage the development of a pungent, red-orange exterior mold and a pungent, earthy flavor. Examples are Epoisses, Reblochon. More widely known as "the stinky cheeses."

Pressed / Firm - These cheeses can be "cooked" (the curds heated before cheese making, giving a sweet, nutty, cooked-milk flavor [think Gruyère]), or not (think Cheddar). Moisture is removed from the curds by applying pressure to the cheeses. This results in a wide range of semi-firm to very hard cheeses, additional hardness coming from age and the further dissipation of moisture. Includes everything from Emmental ("Swiss cheese") to Cheddar to Manchego to Tommes.

Blue - Blue mold is added to cheese paste. The cheeses are then formed, and then pierced with something very long, thin, and sharp to allow air to get to the interior mold. If this step is not performed, you'll get a "white blue," a blue cheese whose blue mold was never given the air to develop.

Processed cheeses - Mixing stuff together in less traditional ways, often in a factory. Think Kraft singles.

Other characteristics may be added to the cheeses in these categories according to their recipes (like the "holes" in Emmental, or the yellow-orange color of Cheddar, Mimolette, or some Goudas), or may spontaneously occur due to the circumstances of their handling (like the occasional ribbons of blue mold that adorn traditional English cheddars).

Answer 2

hard cheese ,semi, and soft Parmesan reggiano, mozzarella, and brie