A species is a group of organisms that are able to interbreed successfully, producing offspring who themselves can reproduce. When one group of organisms can no longer breed successfully with another group, then those two groups are considered to be different species. There are several reasons that prevent successful breeding, but one condition must be met before those barriers can be erected, Reproductive isolation.
Reproductive isolation describes a condition where two populations are isolated from each other, so they can not meet to breed. The most common situation is geographical isolation. In this case, some physical barrier has split a small population from its parent population. When separated for a long period of time, the two populations can diverge from each other either physically or behaviorally. When this has happened, then they may not be able to reproduce with each other when the barrier has been removed. This specific example is termed allopatric speciation, where a geographic barrier imposed reproductive isolation, thus leading the the evolution of two new species.
Another form of speciation is called adaptive radiation. In adaptive radiation, a species enters a series of different habitats, where open niches exists. Because the new habitat differs from the earlier habitat, the species face a form of directional selection, where one extreme trait is being selected for. These events leads to the evolution of a new species.
Does natural selection lead to new species. Many argue that new species have never been seen, but a recent example puts weight into Darwin's theory. In the Himalayan mountain range, a group of songbirds and salamanders have experienced geographical isolation, which lead to allopatric speciation ("Evolving Before Our Eyes" by David Perlman, 2001). In this case, theses warblers have been separated from each other by changes in the mountain range. During the years of separation, their songs that they use to entice mates have shifted. So they do not recognize each other as potential mates. This is an example of a prezygotic reproductive barrier, where two groups are prevented from mating.
Postzygotic reproductive barriers were known to early European settlers of the Americas. A mule is the result of a horse and a donkey mating. Yes, horses and donkeys can have offspring, but the offspring, the mule, is infertile. Mules can not have offspring. This breaks the definition of species, showing the horses and donkeys are closely related, but are in fact different species.