The digestion ends in the small intestine. Some people assume that it ends in the anus but it is the anus from where the digested food is removed. The small intestine is in charge of breaking down some insoluble molecules to make sure that the body gets enough nutrients from the food.
The large intestine will only get rid of the excess water to make it easier for feces to be formed and removed through the anus. Digestion starts in the mouth with the help of the saliva. The food will go through the various digestive parts of the body and it will end when the food goes out of the small intestine.
Most of the food is digested even before the food reaches the small intestine. Digestion begins in the mouth with the salivary amylase enzyme. Stomach mixes up the food and turns it into liquid form. The majority of food is hence already digested. Since Digestion is breaking up of insoluble food nutrients into soluble food for absorption, the ending point of digestion can be labeled as the stomach. But a small part of the small intestine, namely the duodenum, is the input point for pancreatic and liver fluids which breaks down targeted insoluble molecules. This is the reason that digestion finishes in the small intestine. The large intestine's only job is to absorb excess water to convert the digested food into feces for excretion hence it is left out of this argument.