Rennet is an extract from the fourth stomach of young ruminants, such as cows, goats, and sheep. It contains a number of enzymes that are designed to help these animals digest their mother's milk, and when added to milk, it will cause the milk to coagulate, forming the curds and whey that are so essential in the cheesemaking process. Humans have been working with the extract for thousands of years, and it is typically readily available in stores that carry cheesemaking supplies; it can also be made at home, if you happen to have access to the necessary ingredients. For vegans and kosher Jews, non-animal alternatives to rennet are available.
There is a great deal of mythology surrounding the history of cheesemaking, because humans have been making it for a very long time, and the steps involved are actually fairly complicated. The stomachs of ruminants have historically been used to make bags and sacks, and food historians theorize that someone must have stored milk in one a bit too long, allowing it to curdle, and the curdled milk was then turned into a food product. Modern rennet is created through an extraction process that yields neat, dry tablets or a liquid that is very easy to work with.
Traditional rennet was made by washing the stomach of a young ruminant after it has been slaughtered, and then salting it. The salted stomach is kept in dried form, with cooks snipping off small pieces and soaking them in water when they have a need for the extract. Some cheesemakers continue to make and use it in this way, but the vast majority use commercially processed rennet, which is made by creating a slurry and then subjecting it to a compound that will cause the enzymes to precipitate out.
The main enzyme in the extract is rennin, although there are a few others, and the precise content depends on the animal the it comes from; sheep rennet, for example, is different from that from a cow. When added to milk, the enzyme causes the milk to coagulate, essentially starting the digestion process. Once curds have formed, cheesemakers can cut them, drain them, and pack them into molds to make cheese.
Several plants produce natural rennet compounds, as do some microbes, and these non-animal sources can be found for sale in stores that cater to vegans, and in shops that produce kosher dairy products. Vegans should be aware that cheesemakers are not required to disclose the source of their rennet, so unless a cheese is specifically labeled as vegan, it may contain extracts from an animal. Under Jewish dietary laws, milk and meat cannot be mixed together, so cheese that is certified as kosher or pareve will not contain the extract. It is also possible to find cheeses that have been coagulated with acids like lemon juice; paneer is one such cheese.