The biological species concept gives an explanation of how species form (speciation). A biological species is a group of individuals that can breed together (panmixia). However, they cannot breed with other groups. In other words, the group is reproductively isolated from other groups.
"The words 'reproductively isolated' are the key words of the biological species definition". Ernst Mayr.p273

According to Ernst Mayr, a new species forms when an existing species splits. A similar idea had been suggested in the 19th century by Moritz Wagner. Dobzhansky described the role of reproductive isolation in the formation of new species. Once a species lives in two different areas, the geographical isolation makes breeding between the groups reduce or stop. Each group develops features which make breeding between them work less well. Eventually, each group becomes a 'good' biological species, because the two species do not reproduce with each other even when they are together.

This is still considered the most common reason for species splitting, and has the technical name of allopatric speciation. It is contrasted with sympatric speciation where speciation takes place even though all the members live in the same area.

Its role in taxonomy 
The biological species is the main reason for classifying living things, but applying it in practice is not easy. Good examples of this are the following:

"Lack of mating, sterility or inviability in inter-strain crosses has never been taken on its own as good evidence of separate species, including even Dobzhansky's own studies of sibling Drosophila species".

"The biological species concept has no bearing on asexual organisms, and has severe limitations in its application to other organisms. Endless time and effort can be expended on discussing the species concept, and devising yet further modifications and variations on proposed specific definitions. Such debates have great merit in encouraging more objective reasoning about evolution and taxonomy but, in the meantime, human predation and destruction of habitat continue to erase the subject species. Time is against us... Different species concepts produce [groups of subspecies], so it is futile to contemplate a definitive taxonomic system".

What this is saying is that, in practice, the biological species concept is not sufficient by itself to decide on the classification of animals.

Historical developments

John Ray 
In 1686 John Ray introduced a non-evolutionary biological concept. To him, species were distinguished by always producing the same species, and this was fixed and permanent, though considerable variation was possible within a species.

The idea of a species as a physical type of organism had a long history. This survives as the concept of a type specimen in taxonomy. It was the way Linnaeus worked in his binomial classification, and is still useful for amateur naturalists today.

Charles Darwin 
In the Origin, Charles Darwin said that species were labels which experts gave on the basis of their observations.
"... I look at the term species as one arbitrarily given for the sake of convenience to a set of individuals closely resembling each other...".

But twenty years before, he had a much better idea. He thought of species as maintained by reproductive isolation. He even says "Hence species may be good ones and differ scarcely at all in any external character". Here he quotes the two sibling species of leaf warbler discovered in England by Gilbert White in 1768. At this early stage in his career, Darwin came very close to the modern biological species concept.p266

Modern era 
In the last 70 years two ideas have dominated the way professional biologists think about species.

The first is the population genetics concept. This is where a species is seen as a group which can mate together even though they are all to some extent different. It amounts to saying that a species is a gene pool.

The second is the use of DNA sequence analysis to show whether similar-looking species are genetically different from each other. This is especially useful when it is not practical to do breeding experiments.

Sibling species 
Sibling species are often called cryptic (hidden) species because their differences can only be seen by analysing their DNA. They are very common in the marine environment.

Many cryptic species exist in all habitats. In the marine bryozoan Celleporella hyalina, DNA sequence analysis was used to show that more than ten ecologically distinct species that had been diverging for many millions of years.

Evidence from the identification of cryptic species means that older estimates of global species richness are too low. For example, mitochondrial DNA research suggests there are at least 11 genetically distinct populations of giraffes. Similarly, the Amazonian frog Eleutherodactylus ockendeni is at least three different species which diverged over 5 million years ago.

References 

Evolutionary biology