In 1859, Charles Darwin published his most famous work, On the Origin of Species. In it, he introduced a theory to explain how evolution occurs: adaptation by natural selection. Since then, especially after additional advances of scientific understanding in the 1930s and 1940s, natural selection has gone on to become the centerpiece of modern evolutionary theory and the foundational basis of all modern biology.
So what is natural selection? To understand how it works, we must first understand evolution. The basic concept is very simple: evolution is any change in a population of living things over time. The small changes of microevolution are easy to see, for example northern England peppered moths which started out light-colored on trees with light bark, but became dark in areas of trees covered with soot. The larger changes of macroevolution, like speciation, simply extrapolate as very long series of small changes over much longer timeframes.
Natural selection explains how and why these changes occur. In any population in a particular environment at a given time, some individuals will be better adapted to the environment than others. According to natural selection, these individuals will be more successful at passing on their genes to the next generation. Other individuals tend to die out before reproducing. After a number of generations, most individuals will have the successful genes.
Of course, then the environment changes, which may cause the previously successful genes to stop being so successful - there is no absolute "best" in evolution because "best" depends on whatever the environment is at the time. Also, new variations continually arise in the population due to mutation - which in turn makes some individuals better adapted to the new environment than others. Which in turn leads to more natural selection.
Those are the barebones basics, anyway. For more details, read on through the articles below.