Normal, non-vegetarian cheese is made with
rennet, which is used to
curdle the milk. Rennet is made from the enzymes of a
calf's stomach, which makes it decidedly unvegetarian.
To make vegetarian cheese, either artificial rennet (a bacterial culture) or no rennet is used.
Artificial rennet is also used for kosher cheese, because kashrut forbids the use of a meat product (rennet) in a dairy product. For true kosher cheese however, more is needed than just artificial rennet: the milk that is used to make the cheese as well as all other ingredients should also be kosher.
See rennet or cheese for much more on this subject.