Charles Darwin's theories of evolution and of natural selection have been and continue to be attacked on two main fronts. One of those fronts involves individuals who object to the notion of "survival of the fittest," a phrase often attributed to Darwin but more accurately linked to Herbert Spencer, a 19th century biologist influenced by Darwin's On the Origin of Species. The suggestion that the fittest somehow have an intrinsic right to conquer and subjugate weaker beings is anathema to many social liberals whose views, influenced by Thomas Jefferson among others, argue that the weakest among us have rights equal to the strongest. Equality under the law, after all, is the bedrock of democratic governance. That the fittest subjugate the weakest, however, is a product of objective observations of the natural world, and the entire point of human beings occupying the top rung of the evolutionary ladder is that we are capable of ordering society according to principles of fairness and equality.
The second, more enduring attack upon Darwin's theories, especially those developed in The Descent of Man, involves the age-old struggle between those who subscribe solely to Biblical passages regarding the creation of man as described in the Book of Genesis (creationism) and those who accept scientifically-derived conclusions regarding the evolution of human beings from other species, specifically, from apes. Those debates remain unresolved for many, and have a long history. The 1860 debate between Darwin-supporter Thomas Henry Huxley and religious figure Bishop Samuel Wilberforce at the Oxford University Museum of Natural History (see: http://www.oum.ox.ac.uk/learning/htmls/debate.htm) was an early and important public airing of both sides' positions. More well-known, especially in the United States, was the Scopes Monkey Trial of 1925, in which a Tennessee High School teacher, John Thomas Scopes, was charged with the crime of teaching evolution, which, by definition under the terms of the state law, was a challenge to the notion of Divine Creation. Scopes was famously defended by the renown defense attorney Clarence Darrow, while the state's case was argued by the equally prominent Williams Jenning Bryan, a fierce advocate of legally proscribing the teaching of evolution in public schools.
In short, then, Darwin's basic argument was that man evolved from other species, which was strenuously opposed by fervent believers in the notion of Divine Creation. Man, according to the latter, was created in God's image, and did not evolve from a lower form of life.