Chewing the cud is a method of eating preffered mostly by large herbivorous mammals.

For this example we will look at the eating habits of the common cow. Cows graze for roughly eight hours a day at a rate of roughly 890 bites per hour. Their main source of food is grass. Grass is very high in cellulose and difficult to break down. So when cows eat, they tear away the vegetation with their specially adapted mouths, then straight away they swallow it into a seperate compartment in the stomach, which is also the largest, called the rumen.

Within the rumen there are special micro-organisms and bacteria that have evolved to aid in the breakdown of the thick cellulose walls of the swallowed vegetation. The food will remain in the rumen until the cow has finished grazing.

The cow will then regurgitate the semi-digested food and spend a further 12 hours chewing it with it's specially adapted rear teeth. These teeth are called hypodont, meaning that they do not wear away evenly, they are rough and dented. This aids in the breakdown of the tough cellulose wall of the vegetation, and when the regurgitated food is broken down enough, the cow will swallow the food one more time to it's other stomachs, where it's body's natural digestive system will now be able to finish the job. This is what is known as chewing the cud.

Other animals that chew the cud (ruminants) would be gazelles, giraffes, moose, antelope, caribou, sheep, goats, and deer.

Other animals such as kangaroos and rabbits have digestive systems similar to ruminants, but this if for another writeup.

Info from various agricultural and scientific sites.