"The Selfish Gene" is explicitly directed at the layman, and absolutely no knowledge of biology is assumed. While this presents a danger of boring readers (such as myself) who are already familiar with DNA and meiosis, the colorful metaphors Dawkins uses throughout the book do much to keep the reading engrossing and entertaining.
After a lengthy exploration of basic biology, covering topics such as DNA and the origin of life, Dawkins introduces the gene-centered view of evolution that has long been textbook orthodoxy. Dawkins uses the remainder of the book to look at various types of animal behavior in an effort to convey some general conclusions and tools to help the reader understand evolution and natural selection. Much of his effort is devoted to explaining behavior in terms of the 'selfish gene' - especially social behavior that has long been held to have evolved 'for the good of the species.' Dawkins shows that how fundamental axiom of natural selection (that the genes best at surviving and reproducing will eventually spread through the gene pool) leads directly to the selfish gene and the behavior exhibited by nearly all animals (humans being the prime exception).
I should warn that conservatives would probably not enjoy the book nearly as much as I did. Dawkins is an open secular humanist with socialist leanings, and is not worried about offending the delicate sensibilities of creationists and fundamentalists. This book should only be read by those willing to 'accept' the validity of natural selection and evolution; others would only waste their time. I would direct readers seeking a more scientific discussion of these issues to G. C. Williams's "Adaptation and Natural Selection." All others will most likely enjoy "The Selfish Gene" a great deal and finish the book with a new appreciation for and understanding of evolution and biology.