The definition of species is a somewhat contentious issue. The most widely
accepted definition is the biological species concept, which says that
interbreeding populations are within one species. This does not easily
apply to asexually reproducing forms, must rely on assumptions in dealing
with extinct forms, and requires judgement as to how much hybridization is
too much to be interspecies, but seems to be the best available definition.
Using this definition, many species have been and continue to be created by
hybridization. If the hybrid is viable and able to recognize the
chromosomes of each parent as distinct sets (rather than pairing with each
other), then it will have twice as many (or more, in polyploids)
chromosomes as either (or any) parent species. This will generally prevent
interbreeding with the parent species. However, if the hybrid is able to
reproduce (either asexually, by self-fertilization, or with other hybrids,
possibly asexual clones of itself), it can establish a new population of
individuals reproductively isolated from each other. This happens
frequently in plants and certain lizards, and at least rarely in other
groups. Thus, we have seen new species (and even new genera, by
hybridizing representatives of different genera) formed in this manner.
Probably more common, and certainly more common in animals, is a more
gradual separation of species. This is especially likely after two parts
of a species are separated by some boundary (rivers, mountains, deserts, or
whatever else is difficult for the species to cross). Because it is
gradual, we have difficulty observing this in action, except in things that
reproduce very rapidly. Bacteria can differentiate into different sorts in
lab experiments, but as asexually reproducing forms with few obvious
features, the definition of bacterial species is challenging. Some flies
have differentiated into reproductively isolated populations, thought to
represent new species, by specializing on different kinds of fruit. As
they mate on the fruit, an apple-preferring female will not meet a
plum-preferring male. For vertebrates, however, such change takes longer.
We can see species in different stages of change from a single species to
multiple species. A readily accessible example can be found by comparing
and older and a newer edition of a field guide to birds. Some species
separated in the old one will be lumped in the new one and vice versa.
This is because further research suggests that some populations, though
noticeably different in appearance, still interbreed at a moderate level.
Others, though similar, turn out to not interbreed. Another example comes
from a fish in Iceland, Arctic char if I remember correctly. A
recently-formed volcanic lake now contains four distinct populations of the
fish. Some have specialized for bottom-feeding, some for open water, and
different sizes of prey. They also have different preferences for breeding
locations. In the unlikely event that the lake were to last long enough
(glaciation or volcanism is likely to come first), these populations could
develop into full species.