While this concept is fascinating (and takes us through most of the book with an eloquent explication of evolution), Dawkins manages to sneak in a wammy with his final chapter (in later editions of the book, other chapters follow). He posits the construct of a "meme", the analogous mental equivelent to the physical gene.
The meme, Dawkins says, can be a commercial jingle, a popular phrase, an idea, an image -- it's kind of an atomic unit of thought, if you'll forgive the gross generalization. It's cliches, it's inside jokes, it's the second phrase of the topic sentence of the fourth paragraph Douglas R. Hofstadter ever wrote.
Dawkins suggests that memes try to propogate, just as genes do. However, the medium in which the meme propogates is in the human consciousness. What's the measure of success for a meme? If you remember it long enough to transmit it to someone else, then it's viable. Evolution and selection take place, as well -- simply consider the accumulated body of knowledge of the human race as representing our "meme pool", just as the accumulated body of bodies of the human race represents our gene pool.
The kicker here is the time frames. Genetic evolution takes place at difficult-to-conceive-of rates, over the course of many generations and thousands of years. Memetic evolution, on the other hand, takes place over the course of years, days, minutes, or seconds (especially with communications infrastructures like the Internet!).
In a sense, our memetic evolution has almost recapitulated our genetic evolution. Once we've mapped our genome, and can manipulate it, then it will be our ideas -- our memes -- that dictate our bodies -- our genes. At that point, evolution can race onward at the speed of though into the wildest imaginings of our kind. Scary, huh? And our memetic successors may well be the eventual potential fruits of current Articial Intelligence and Artificial Life research. We'll likely not even have time to biologically evolve our own successors before we engineer them memetically.
James "Eric" Tilton, Person Thinking About Cognitive Science, firstname.lastname@example.org