A pedigree is a dog's family tree. It lists the registered names of the dog's parents, grandparents, great-grandparents (etc.) of the dog - most printed pedigrees go to 4 or 5 generations. Other information may or may not be included, such as date of birth, color, registration number, call name, or titles earned. Titles earned are almost always included - Championship, Sire or Dam of Merit, Obedience or Agility titles.
A printed pedigree is not needed to show a dog - to be eligible for an AKC show, a dog must be registered, so the parents (and thus the entire pedigree) are already on file.
Pedigrees are useful when planning a breeding program. Knowing your dog's ancestry, and the ancestry of the dog you are considering breeding to, is necessary to help you determine how closely related the dogs are. Sometimes dogs that seem unrelated in the first few generations end up with the same dog numerous times in the fourth or fifth. Knowing how many Champions or other title-holders are in a dog's pedigree will tell you about the line, and it's tendency to meet the breed standard in form or function.
Specific information about the ancestors in your dog's pedigree is especially helpful (and sometimes hard to come by). Knowing when the dogs died, and of what, can help you make decisions on health standards. Temperament and health are also vital factors in a breeding decision. Knowing the history of these things in both dogs' lines can make the difference between a good breeding and a great one.
A pictorial pedigree can sometimes be especially helpful - while there's no guarantee that a dog will sire puppies that look like him, consistency of type throughout a pedigree will increase your chances of getting that type. Seeing other offspring of the dog you're looking at - puppies and adults - can be helpful, too. Many dogs who are lacking in a certain quality (say, rear angulation) produce pups who possess the quality. The best breeding results in a puppy that is better than his parents, in keeping with the goal of continually improving the breed.