Food production has become highly complex. Many ingredients are used in seemingly simply food products that are not suitable for vegans or vegetarians. For example, gelatin(e), an ingredient often used in many products including candies and yogurts, is usually made from the remnants of cattle, pigs or horses.
Tallow, which is often used as a lubricant, is a rendered form of beef or mutton fat. Cheese contains rennet, a coagulant, which is often made from pieces of the stomach lining of certain animals. Even carmine/cochineal, which is used as a dye, is actually crushed up beetles (yes, you read that correctly).