The following material is flagged Green Level. It is intended to reflect material that the author believes to be a matter of consensus among experts in the field. This belief may be incorrect, however; and as the author is not an expert and does not have an expert fact-checking the article, errors may creep in.The Basics
First, let me explain what the theory of evolution actually says, since this is the most common point of disagreement. There are things (such as living beings, although a semi-controversial application of the theory called memetics posits that this can be expanded to include abstract information as well, and a type of robot called a von Neumann machine that constructs copies of itself would also count) that can "copy" themselves, which we will call replicators. Sometimes, when a replicator makes a copy of itself, it makes a small mistake, called a mutation. Sometimes, this mutation has no effect beyond possibly changing the effects of future mutations, such as a mutation in a set of "junk DNA" or in the comments of a piece of code. Sometimes, the mutation is neither harmful nor helpful, such as a change removing a vestigial organ. Sometimes, the mutation is harmful, such as a vertebrate forming without a DNA sequence coding for brain tissue. But sometimes, once in a great while, the mutation is actually useful, such as increasing how well the organism can process food.
Now, suppose that our mutant is in a place where its mutation is useful. Suppose that we're talking about a bear in a cold region that has fur better able to trap heat, or the like. That bear is going to find it a little bit easier to not freeze to death, and as such will have a slightly better chance of leaving copies of itself in the next generation of bears.
(As an aside, it is important to point out that a beneficial mutation is one that is beneficial in that specific environment. A bear with heat-trapping fur is at a disadvantage if taken to a place hot enough that water boils quickly, just as a heat-repelling one is at a disadvantage if taken to a place cold enough that all water is frozen. This will be important later on, when we talk about speciation. It will also be on the test, so I do hope you are taking notes.)
Now then, we have a bear in generation X with a 0.1% better ability to handle cold environments. This bear has a copy in generation X+1. Now, let us look at what that copy can do. Since it is a copy of the original bear, it also shares that 0.1% better ability. And if that bear manages to put copies in generation X+2, they will also share the improved ability to handle the cold, and so on. And eventually, those small chances add up. If in generation X+2 this improved ability saves one of the mutant bears, that means that there will be more of the bears with this ability in generation X+3. And this continues for a while.
But there is another factor: cold is not the only threat to the bears. Hunger is also a problem. Eventually, the bears hit a limit on the number that their food supply can support. With this limit in place, and the steadily growing number of bears able to handle cold, eventually the entire population of bears becomes able to handle the cold.
(Note that simple probability indicates that eventually, the bears with the higher survival rate would displace the bears with the lower survival rate, even if there were no population cap. This takes even longer, though.)
Now, let us assume that a second mutation shows up. This mutation repurposes the protein responsible for fur that grants resistance to cold so that when expressed in fur, it will still have its original purpose, but when expressed in blood, it will grant further cold resistance by the same amount. (Yes, I know that this specific for-instance has an incredibly low chance of actually making any kind of biological sense. But we're discussing hypothetical abstractions anyway, and it gets the point across.) So, the two mutations spread through the population at about the same rate, right?
Wrong. Remember that the second mutation works by repurposing the first. In bears that do not have heat-trapping fur, the mutation for cold-resisting blood will, at best, do nothing. At worst, it will repurpose some other protein, and do something very interesting to the bear's internal biochemistry. So, the cold-resisting-blood mutation can at best spread through the population at the same rate as one with a neutral effect, and at worst cannot spread beyond the subset of the population that has the heat-trapping-fur mutation. In other words, the mutation is of no beneficial effect without the one it modifies.
Next, suppose that, once the cold-resisting-blood mutation has become universal (and of course, the heat-trapping-fur mutation with it), a mutation happens in the heat-trapping-fur gene that causes it to function better, but only if the cold-resisting-blood gene is also present. (And at this point, the specific examples start approaching impossibility, but again: for-instances, hypothetical abstractions, gets the point across, et cetera.) Since the cold-resisting-blood gene is universal, the heat-trapping-fur-mark-2 mutation has no drawbacks. Since a heat-trapping-fur gene still exists (and can still be adapted for the purposes for which the cold-resisting-blood gene uses it), the system still works. And so, we have a system that is stable as it is, but which will fall apart if any component is removed.
Now, let us suppose that the bears somehow split into two populations. Maybe the ice floes they live on are broken and begin drifting apart farther than the bears can swim. After millions of millions of generations, the bears build up a large number of mutations. Now, suppose that one mutation in one population is incompatible with another mutation in the other population. If those two mutations occur in the same individual, that individual will die off.
Contrary to how it may appear, this is not a disadvantage. As was mentioned above, a trait does not have to be beneficial in all situations in order to be beneficial; it only must be beneficial in the current situation. Similarly, it does not have to avoid being harmful in all situations in order to be harmless; it only must be harmless in the current situation. And the inability to breed with an individual one could not otherwise breed with is not a harmful trait. If the trait is beneficial (or even just allowed to spread over a few thousand more generations or so), the bears on the two floes will be separate species.
This will be continued later, after I have received some feedback on it. For now, this should cover the basics.
EDIT: I realized that I need to cover some more material, such as the evolution of cooperation. This will be discussed later.