Natural selection is a statistical premise put forth by Charles Darwin in The Origin of Species, a work that relies so heavily on natural selection that its full title is actually On the Origin of Species by Means of Natural Selection. Darwin's thesis relied on the writings of Thomas Malthus, who claimed that human populations would grow faster than their ability to feed themselves.
Applying Malthus's ideas, Darwin realized that organisms throughout the entire living world produce more children than can possibly survive, and yet their populations remain stable. Working before Mendelian theory had provided a coherent view of inheritance, Darwin concluded that if organisms introduce variations, then it would inevitably follow that only the most fit for survival and reproduction would pass on their characteristics to progeny, thus facilitating species evolution via a natural process.
Natural selection lies in contrast to artificial selection, in which breeders actively select desirable traits and only allow organisms possessing those traits to reproduce. Darwin had noticed how quickly artificial selection works (it had created a huge variety of dog breeds on the planet in just a few thousand years of selective breeding), and he concluded that nature would likely operate by a slower mechanism that would nevertheless achieve large-scale change over eons.
Of course, natural selection does not offer any mechanism by which new variations can be introduced; Darwin assumed that variation somehow occurs spontaneously in each generation. Without a modern understanding of genetics, he could not realize that variation is ultimately attributable to genetic mistakes (i.e., mutations).