We hate to ever have to share bad news with you guys, but we have some today: not all cheese is vegetarian. Although milk is an animal product that can be harvested without harm to the animal, there is another element in the cheese-making process that can't be -- rennet. Rennet is the coagulant that turns milk into cheese, therefore improving it by roughly 600 percent (it should go without saying that that is not scientific data). While there are microbial and vegetable-based rennets, many cheeses are made with animal rennet, which is an enzyme harvested from the stomach of ruminant animals like cows and goats. Since there isn't a way to harvest animal rennet without first slaughtering the animal, those vegetarians who choose not to eat meat for ethical reasons would absolutely not want to ingest this stuff.
The use of animal rennet is widespread in the cheese-making community, especially in European and Old-World style cheeses. For cheeses like Parmigiano-Reggiano that are D.O. (designation of origin) protected, cheese-makers have to use an exact method and exact ingredients in order to legally be called by that name. But don't despair, although the cheeses on this list are traditionally not vegetarian, you can find versions of them that use vegetable or microbial-based rennet, although they will have to be called something like vegetarian Parmesan or Parmesan-style cheese.
If you are vegetarian, or considering adopting a vegetarian diet, these are cheeses you should definitely watch out for. Although there are vegetarian versions of these cheeses made in some places, for you, it might be better to be safe than sorry, and always check the ingredients, if they're available to you. In a cheese shop, anyone behind the counter worth their salt will be able to tell you what kind of rennet was made in the cheese's production. This is by no means a comprehensive list, but these are some of the most popular and familiar cheeses that are usually not vegetarian.