Darwin pretty much sums up natural selection in the beginning of the third chapter, "Owing to this struggle for life, any variation, however slight and from whatever cause proceeding, if it be in any degree profitable to an individual of any species, in its infinitely complex relations to other organic beings to external nature, will tend to the preservation of that individual, and will generally be inherited by its offspring. The offspring, also, will thus have a better chance of surviving, for, of the many individuals of any species which are periodically born, but a small number can survive," (Darwin, 132).
Darwin preferred the view of competition between individuals over cooperation in groups as the main drive of evolution, a debate that is still strong today -- how much of a species survival is determined by individual success against group success? Social Darwinism is all about competition, and not really at all about biology at all. However, it is an important distinction to make. Survival of the fittest is Social Darwinism, actual Darwinism is more along the lines of the fittest having the greatest reproductive success. Social Darwinism came about around the time of Darwin's publication, and was quite influential in an already competitive Victorian society. Darwin himself didn't actually espouse the ideas, they were actually started by Herbert Spenser. Social Darwinism was used as a justification for imperialism, exploitation, unchecked capitalism, eugenics, and a slew of other evils (2). The point of this little tangent is that Social Darwinism does not equal Darwinism, at all. I can't tell you how many times I've seen this mistake propagated by the media, specifically network news.
Back to the book, chapter 3 is also kind of short and is all about natural struggle as well as geometric increases in population.
Geometric increase is basically exponential increase, meaning that the population will increase steadily exponentially rather than at a steady linear rate. This accounts for rapid increases in populations. Darwin used the example of elephants, because they live a long time and produce a very small number of progeny. His math leaves a lot to be desired, but he was right in a very basic sense. So Darwin proposed that an elephant lives about 90 years and has 3 calves, and each of those calves grow up to have 3 more. Darwin estimated that after 500 years, there would be something on the order of 15 million elephants stemming from one pair (1). Well, no, actually if we're talking approximately 5 generations, that would be 3^5 elephants, or 243. This is opposed to linear growth in which 5 generations would yield 3x5 or 15 elephants, big difference. Did no one think to check these calculations before publishing; I mean, if Darwin had been right, we'd be up to the empire state building in elephants. Also, not every one of those 3 calves are going to necessarily have three calves because some of them are going to have to be male (or else there will be no elephants). Regardless, geometric growth is how species propagate.
Speaking of those elephants, not all 3 calves will survive to produce calves of their own, thus the struggle for existence. "Hence, as more individuals are produced than can possibly survive, there must in every case be a struggle for existence, either one individual with another of the same species or with the individuals of distinct species, or with the physical conditions of life," (Darwin, 134). Basically, more progeny must be produced because not all of them are going to make it. This also brings up differences in parental investment, species that invest a great deal in their offspring generally have fewer than those who do not beyond birth, but I'll save that for a separate blog post.
Darwin brings up the effect of climate on the struggle for existence, and how particular occurrences such as cold winters or dry summers impact populations short term. Later in the chapter Darwin suggests that certain varieties could mix to produce an individual better suited to a certain environment, and varieties ill-suited to the climate will soon disappear (1). This is basically a primer for natural selection, which is detailed in the next chapter.
He also notes the impact of one species on another. Darwin recognized the relationships between species, and used an example of certain bees associated with specific flowers. Darwin extends this to the bees being impacted by field mice, which are impacted by cats. Therefore, one could muse that a decrease in cats would lead to a decrease in flowers (since the mice would increase and damage the bees in greater number) (1). This co-evolution is widely seen today, and even in ourselves. E. coli is one such example; it has evolved to live in our guts and we depend on it to aid in our digestion.
Darwin leaves us with this small comfort, "When we reflect on this struggle, we may console ourselves with the full belief, that the war of nature is not incessant, that no fear is felt, that death is generally prompt, and that the vigorous, the healthy, and the happy survive and multiply," (Darwin, 143). Doesn't that just leave you feeling warm and fuzzy?
 Darwin. On the Origin of Species, 2nd ed.
 Social Darwinism. ThinkQuest. 2000. http://library.thinkquest.org/C004367/eh4.shtml