Social Darwinism is a 19th-Century philosophy which applied the concept of "survival of the fittest" to human society. Though there are other variants of social Darwinism (including Nazi-style eugenics), the idea is primarily associated with an argument for laissez-faire capitalism put forth most famously by Herbert Spencer.
The most basic form of social Darwinian "defense" of capitalism argues that mankind, like all other species, evolves by the process of competition for scarce resources. The most "fit" — i.e., the ones best able to gain and utilize natural resources — survive and reproduce, while those less capable eventually die off and are removed from the gene pool. Laissez-faire capitalism allows the fittest (the Rockefellers and Carnegies of the world) to gobble up all the resources and reproduce, while the unfit (such as the handicapped, the mentally retarded, or the lazy) die off and fail to reproduce. Any interference with laissez-faire capitalism (such as social welfare programs or regulations on business) weakens or destroys the gene pool.
This argument, as a form of pure biological determinism, utterly ignores the role of the human mind. While it recognizes the importance of "intelligence," it holds intelligence as a purely biological, innate attribute. It doesn't take too much of a sophisticate to see that, if you somehow killed off every lazy person in the current generation, next generation you'd have a whole new crop of them — because laziness is not a product of genetics.
A somewhat more sophisticated version, exemplified by the stories of Horatio Alger, gives notional credence to the mind. It holds that the survival and thriving of the fittest in any field serves as an example, while the failure or death of the unfit serves as a cautionary tale. While significantly better than crude form of social Darwinism, it is still premised on the idea that capitalism requires mass death, and it is still based on the altruistic premise that the purpose of ethics/politics is the betterment of society.
Note that social Darwinism in either form is not properly a defense of capitalism, but a critique of everything else. It doesn't say that capitalism is moral, or just, or based on objective values, or rooted in unalienable rights. It carries only the grim message that "if you do anything else, the race will die in misery." (In point of fact, most social Darwinians believe that extinction is inevitable anyway, as human beings will eventually use up all the resources in the environment. Thomas Malthus, though writing before Darwin, made the classic argument for this view which was later absorbed into social Darwinism.)
Although, like Ayn Rand's ethics, this argument claims to be rooted in biology, it is rooted in precisely those elements of biology that are least-relevant to mankind. As Stellavision noted, human beings do not survive by cutthroat competition for scarce resources. Human beings thrive by productive creation and mutually-beneficial trade of increasingly-abundant values.
Imagine a young man trying to sort out his political opinions, as most people do in their teens and early twenties. On one hand, there is a stern-faced exponent of the view that mankind, while eventually doomed to miserable death, can stave it off for a time only by accepting mass death of the unfit while the fat-cats born with unfair advantages exploit us all and live in opulence. On the other hand, there is a Marxist/socialist who promises a future of solidarity and prosperity for all, if we only expropriate the wealth from the few and distribute it "fairly" to the masses. Is it any wonder that social Darwinism has led far more people to reject capitalism than to accept it?
As for the question why people associate Objectivism with social Darwinism, the obvious answer is that those on the Left have taken social Darwinism as the straw man, the most obviously repulsive version of pro-capitalist argument, in order to smear the rest of us.
But what about those who aren't virulent anti-capitalists, who nonetheless associate Ayn Rand's ideas with social Darwinism? In most cases, I think it's due to the surface-level association of the fact that Ayn Rand made heroes of businessmen, and so did social Darwinians. The fact that Ayn Rand's businessman heroes produced and spread values, not sacrifice and death, is lost on the casual reader.
Social Darwinism is the application of Darwin's evolutionary theory in a social context. In the wild, animals compete with each other for food, sex, and shelter, and survival of the fittest is the rule. Because animals have no such thing as a concept of rights, they cannot and do not recognize any "right" of other plants and animals to be left alone. Instead, animals compete with each other for scarce resources, often killing each other in the process or dying as an indirect result of this struggle (e.g., inability to obtain shelter could lead to an animal freezing to death).
Two big mistakes people make when comparing laissez-faire capitalism to social Darwinism are: