Natural selection is the proposed mechanism that drives Darwinian evolution. It can be understood as a natural analog to the types of artificial selection methods employed by humans to produce the many different breeds of domesticated animals such as dogs, cats and cattle.
How natural selection works
All organisms in the real world encounter environmental pressures (often referred to as directional selection) that affect fecundity rates. These may take the form of inhospitable climate, scarcity of food or mates, predation by other organisms, accumulation of waste products over time, and the like. All of these pressures make it more difficult to survive and reproduce. Some individuals, however, may be better suited to deal with and overcome such pressures to ultimately reproduce and successfully rear their young. Those that are so suited ("fitter"), pass on their phenotype/genes to the next generation; those that aren't, don't. Over time, the traits associated with more successful reproduction are thus "selected for" and tend to become more widespread in the population.
Darwin is often misinterpreted as proposing that those individuals who survive are the strongest and healthiest ones, an idea summarized in the phrase "survival of the fittest", which is actually due to Herbert Spencer (although Darwin used it, with attribution, in his book The Origin of Species). This idea has been used to reinforce different forms of social and scientific prejudice and is associated with the term Social Darwinism. The misunderstanding is based largely on the semantic ambiguity of the word "fit" (Darwin merely meant being suited to ones environment).