To make it easier to understand Evolution, Dawkins makes an analogy and asks us to imagine a mountain. This is not a simple mountain, it's a mountain of improbability. This mountain has many peaks, some of them very high up and some of them only a few feet above the ground. Evolution is like a mountaineer, trying to get to the highest possible peak. The highest peak represents the best adaptation one could possibly have, while lower peaks are adaptations that aren't that good, but they're still better than nothing.
For example, a finger that has only one bone (and therefore can't be bent to make a fist) is of very little use, but it is better than no finger at all. So a finger with only one bone might be a peak 500m above the ground. A finger with two bones (and all the muscles needed to bend it) could be a peak 1000m above the ground. A finger like ours might be 2500m above the ground.
But there might be many more peaks, for example a finger with ten bones might be only 1500m high, because the finger might break more easily. So more bones in a finger does not necessarily equal being higher up on the mountain.
There might also be very different kinds of hands with different underlying bone and muscle (and tendons, etc.) structures.
So our mountain could have many bases, all of them resulting in a general mountain, of which each has many peaks.
Evolution can only build upon what is already there, so we'd expect the mountain not only to have many different peaks, but also to have one or more completely different mountain bases. Each base represents a different underlying structure.
And now the mountaineer sets off towards the mountain top. But there are two limiting factors:
1) There is fog surrounding the mountain. Our mountaineer can only see the next few steps, but never the peak. So our mountaineer might be lucky and climb the highest peak, but he might also be unlucky (Indeed, the chances are far greater!) and climb a peak that is only a few hundred meters above the ground.
2) The second limitation is that our mountaineer has learned to climb upwards, but not downwards and then re-climb the next peak. What this means in evolutionary terms is that we can evolve something but not de-evolve something. (This is stated by Dollow's Law of Irreversibility) There is one thing that can happen, that isn't an exception to this rule. We can reduce these traits until they are almost invisible, but they will be there nonetheless. These are called Vestigial Organs.
So if we now send off our brave mountaineer, we can make a few predictions. If these predictions turn out to be right, I'd say that that makes a pretty strong case for Evolution.
1) We should expect to have some (most) species evolving imperfect organs.
2) We'd further expect that we can't get rid of those organs and that we can't change them into something else. (For example changing a Human eye into the eye of an Octopus.)
3) We'd expect that closely related species inherit the same traits but more importantly also the same mistakes. Refer to * here.
* Kenneth Miller makes a nice example for this point. In his book „Only a Theory: Evolution and the Battle for America's Soul“ he explains how you could see how one kid is copying from another. The next part is an extract from his book:
Several times in my career as a college teacher I have caught students cheating on exams or term papers. Sometimes it's an easy thing to spot. They pass notes during exams, crib passages from the Internet, or copy from each other's papers. During a final exam in one of my courses several years ago I noticed two students carefully eying each other's work, and watched them long enough to be almost certain that one was copying an essay from the other. When the exam was over I set both papers aside for comparison and confirmed my suspicions. I called both students into my office the next day and told them that the similarities in their essays left no doubt that one had copied from the other.
Once challenged me to back up the charge. „Of course our essays are similar,“ he told me. „We're roommates and we study together. So naturally we gave pretty much the same answer to the same question. That doesn't mean we copied.“
„No,“ I answered. „It doesn't.“ And indeed one of the students had gone to considerable lengths to conceal the similarity of his work to that of his roommate. He had used a different title, had switched the order of several paragraphs, and had clearly rewritten many of the sentences to make them read quite differently. „But I wasn't looking at superficial similarities. I was looking at something deeper,“ I explained, placing the paper in front of him and calling his attention to a series of words, each circled in red ink. „The two of you misspelled the same six words, and you misspelled each of them in exactly the same way.“ Immediately the students realized that I had them.
If the students had worked independently, they might indeed have made similar points in a correct answer to an essay question. They might even have worded a few sentences in the same way. But the notion that the exact same spelling mistakes could crop up, independently, on two different papers was too much to give credence to. There is, after all, just one way to spell a word correctly, but there are an infinite number of ways to get it wrong. And when those errors match perfectly, there can be no doubt that those errors must have a common source. The students pleaded guilty, and threw themselves on the mercy of the court.
The third statement (if correct) should in itself suffice to prove Evolution, but I want to give that topic justice, so I will for now concentrate on points one and two.
The first point will be thoroughly explained in the thread about the eye, but here's a small introduction:
Our eye is far from perfect. We can see quite well, in color even, but we don’t have the brilliant eyes a hawk has. We also have a blind spot, something the octopus for example does not. (That’s because his nerve endings are attached at the back oft he retina, not in front of it, like with us.)
So let us now look at point two. This is what vestigial organs are all about. Here's what a vestigial organ is, as explained by Jerry Coyne in „Why Evolution is true“:
So if evolutionary theory is correct, we'd expect to find characteristics or organs that aren't used to their original purpose anymore.
(...) Evolutionary theory doesn't say that vestigial characters have no function. A trait can be vestigial and functional at the same time. It is vestigial not because it is function-less, but because it no longer performs the function for which it evolved.
One example is the Emu's wings. The Emu doesn't fly with those wings anymore, so they are vestigial, because their ancestors (other birds) did use them for flight. The same is true of Penguins, etc. They are not, however, without function. Emu wings are used to give the Emu balance and to speed it up when running and the Penguin uses its wings for swimming.
The next example is the coccyx of humans.
The coccyx serves no purpose, has the potential of producing a lot of pain and is truly vestigial, if evolution is true. Where did it come from? Well monkeys of course.
The last thing I want to talk about is Atavisms. An Atavism is when an inactive gene is revived. An example of this would once again be the Human coccyx. In very few cases, the genes that express the tail in humans are re-awakened to such a degree that we can see the underlying structure again, so in other words we have an almost complete tail again.
There's really not a lot to be said about this, except that it's exactly what we'd predict, if evolution were indeed true.
I want to end with two things here:
1) If we ever find that there is a vestigial organ/characteristic, but the ancestor species did not have this trait, we'd have to be very skeptical of evolution, if not abandon it entirely. What we'd be looking for is for example a Human having Emu like wings. None of our ancestors had wings so we couldn't possibly have vestigial wings.
2) This poses a huge problem for Creationism/IntelligentDesign. If they are true, we'd expect no relation between humans and monkeys. Indeed, we'd expect no vestigial organs at all, because surely a designer wouldn't build in things like the coccyx, which has the potential to produce great pain. And if there is indeed a use for the coccyx, it would be a very strange fact that monkeys have the real tail and all apes share the coccyx.
I feel like I have left something out at the „climbing mount improbable“ part, so I might have to write an add-on to it.