Rennet is a non-crucial part of the cheesemaking process. It makes cheese coagulate faster, separating the curds from the whey. It is useful when making harder cheeses that need to dry quickly.
The primary enzyme in rennet, chymosin, most often comes from the fourth stomach of a cow. Specifically, it has to come from a newborn calf - calves use it to learn how to digest milk. Occasionally it comes from piglets,who have similar digestive issues. However, there are vegetarian options.
There are two main potential sources for vegetarian rennet: vegetable and microbial. Vegetarian rennet - vegetable enzymes which work in the same way as chymosin - can come from fig leaves, melons, safflower, and wild thistles. Microbial rennet is either fermented from fungi/bacteria or genetically modified; genetically modified rennet is created either by extracting the DNA for chymosin from calf stomach cells, or by bio-synthesizing it without the use of any calf cells.
The Vegetarian Society of the United Kingdom has an information sheet on cheese and rennet, which says that "most widely available vegetarian cheeses are made using rennet produced by fermentation of the fungus Mucor miehei. Vegetarian cheese may also be made using a rennet from the bacteria Bacillus subtilis or Bacillus prodigiosum." They make vegetarian cheese-buying easy for British shoppers by keeping an eye on cheese manufacturers and labelling non-animal-rennet cheeses with a "V." In the United States, such products are often not labeled, but a few suppliers (like Trader Joe's, Whole Foods Market, and other organic-friendly places) make sure to include information on rennet sources on their cheese labels, and sometimes even provide informational pamphlets on the subject. Cheeses are labelled fairly well and frequently in Canada, probably more so than in the United States. Should you reside in another country, YMMV.
Interestingly, vigilant British rennet recognition leads to situations like the one surrounding Walkers Crisps, where many of their meat flavors are vegan (including Barbecue and Beef & Onion) but Cheese & Onion is Not Suitable for Vegetarians because it uses animal rennet.
A writer at vegsource.com's discussion board comments,
"I'm from England, and supermarkets here label vegetarian cheeses with the Vegetarian Society's V. But I also found out recently that almost 95% of cheeses in our supermarkets are made with vegetarian rennet. Supermarkets refuse to label them all vegetarian due to fear of loss of sales from omnivores who don't wish to eat vegetarian food."
There are several political issues affecting the presence of vegetarian rennet in food. On the "pro" side, it prevents nasty surprises for vegetarians who thought they were getting food without meat in it. It also gets points from animal rights for not causing the slaughter of newborn calves, although the dairy industry is not known in general for animal-friendly practices. On the "con" side, people against, for example, the long term consequence of putting fish genes into watermelons might oppose genetically modified rennet; on the other hand, with proper labelling they could stick with vegetable-based rennets, or eventually be able to find cheeses that are labelled "No GMOs!" as some other foods already are.