The digestive system of the elephant is set apart almost exclusively by its sheer size. Apart from this factor, it is very similar to the digestive systems of most other mammals. Likewise, the liver and the pancreas, both key organs that aid in digestion (albeit indirectly), are also much the same as ours, only far larger in scale.
The mouth, where food begins its digestive process, is relatively small compared to the enormous animal itself. In addition, this mouth is not able to open very wide either. Digestion in the mouth is aided by the ample saliva produced by the many salivary glands in the mouth as well as by the mucus glands in the oesophagus. Saliva contains enzymes that begin breaking the food down. It also serves along with the mucus to lubricate the tough vegetation eaten by the elephants, which even includes thorns and bark.
The muscular oesophagus ushers the food into the stomach, which acts more as a storage sac as not much digestion takes place in this organ. The elephant's stomach is cylindrical in shape and the middle of the organ is particularly glandular. From here, food is ushered into the extraordinarily long intestines. The intestine of an elephant can reach up to 19 metres in length!
The intestine is where most of the digestion of the vegetative diet takes place. At the point at which the small intestine meets the large one (or colon), bacteria aids in the fermentative digestion of the cellulose (typical of this diet). This location is called the caecum and is particularly rich in blood vessels. The caecum is divided into many smaller sacs and the products of digestion are absorbed through its relatively thin walls.
Because the elephant only digests and makes good use of 40% of its intake, the intestine is also instrumental in the formation or faeces and the efficient absorption of water. The size of the faeces is often used to determine the age of the elephant as it retains the shape formed by the walls of the rectum, indicating its size.
Because the elephant absorbs so few nutrients from the food it ingests, the dung is rich in nutrients and solid food matter. Therefore, it is beneficial to many other animals, as they are able to feed off of the relatively untouched constituents of the faeces. Common beneficiaries include dung beetles and many species of bird.
An elephant will spend much of the day searching for food to ensure that, although they actually digest relatively little from their food, they are taking in enough to remain healthy. In areas where vegetation is plentiful, the elephant will spend approximately 16 hours a day grazing or eating fruits, nuts and seeds from trees. In areas that are very dry or lacking in healthy plants and grass, the animal may need to spend up to 24 hours a day in search of food. For correct nutrition, the elephant needs to consume between 140kg and 270kg of food every day.
Here is more on Elephants and digestion: http://www.seaworld.org