Most cheeses contain a coagulant (technical term is rennet) that is added to the milk to separate the curds from the whey. Traditional rennet is made from an enzyme found in the stomach lining of suckling calves and lambs. This was likely discovered in the transport of milk in calf stomachs, or when the animals were slaughtered and curds were discovered in their stomachs. In early cheese making, the stomach linings were dried out, ground up, and added directly to the milk. These days animal rennet comes in powder or liquid form, and is highly controlled and sanitary. But the resulting cheese it is technically NOT a vegetarian food.
Thankfully vegetarians need not rule out all cheeses—vegetarian rennet does exist.
The ancient cheese makers of the Iberian Peninsula for example, discovered the coagulant properties of the Cardoon Thistle. And in the wake of mad cow disease, many English cheese makers switched to vegetable rennet, and have yet to switch back. Plant sources of vegetable rennet include Lady’s Bedstraw, Stinging Nettle and Safflower. Microbial (genetically engineered) rennet is also widely available, and is vegetarian-friendly.
Some examples of cheeses made from different rennet:
Animal Rennet: Parmigiano Reggiano, Montgomery’s Cheddar, Gruyère.
Vegetable/Microbial Rennet: Humboldt Fog, Spenwood, Torta del Casar.
No Rennet: Cottage cheese, Mozzarella (most), Queso Blanco.