The basic difference between a prologue and epilogue is that a prologue comes before the rest of the story, and an epilogue comes after. This is a technique that is used in novel writing to provide additional information to the reader, theoretically to increase their understanding of the story, or extend it beyond the time constraints given in the book. Most novels will not include both a prologue and epilogue; one or the other is more common, but of course there are no rules, and some writers will choose to include both if it serves the needs of the work. Often, these will be very short, and will slightly differ in tone from the rest of the story.
A prologue and epilogue are somewhat more common in classic literature than modern literature, though some contemporary novels do include these extra literary techniques. They are written to extend the story, and engage the reader's interest even more. Some writers and editors argue that a prologue and epilogue should not be necessary if the story itself is complete, while others see no problem with them. It is largely a matter of personal and stylistic preference, as well as what a certain publisher is looking for.
As an introduction to a story, a prologue can work very well. It is typically shorter than an actual chapter, and may only be a page or two. Often, it is intended to provide some back story about events that occurred prior to the beginning of the novel. For instance, if a novel is about a family, the prologue might include information about the people who first started that family, or where they first settled. This information might be included in a prologue if it will be relevant to the story later. In fact, if an author takes the time to write a prologue, it is safe to assume this information will be important.
An epilogue provides information about events that took place after the conclusion of the story, sometimes years later. It may detail what happened to certain characters; for instance, if they were children in the book and have now grown up and experienced additional significant events. It may also serve as a teaser for a sequel. Often, an author writing a prologue and epilogue, or one or the other, will write them in a slightly different tone than the rest of the story; for instance, as if he or she is speaking directly to the reader. This helps to separate the prologue or epilogue from the rest of the story and make it more interesting.