Several concepts are recognized, but the biological species concept (Dobshanzky 1937, Mayr 1940) is still the most successful. It states that a species is a group of natural populations that actually or potentially interbreed. As a direct consequence it needs reproductive isolation to maintain a barrier to gene flow between the different species. The application of this concepts is difficult when groups occur in isolated patches and one cannot exclude that they are able to interbreed or when the species is asexual.
Recently, the phylogenetic species concept gains many followers. It recognizes species as the smallest aggregation of populations (sexual) or lineages (asexual) diagnosable by a unique combination of character states in comparable individuals. It has advantages especially when we cannot establish reproductive isolation, e.g. species are extinct, live on different continents.
Recognition of species is sometimes difficult, because the individuals look alike (morphologically or genetically), but as long as species who occur together do not mix they will show some differences that are spread over their whole genome.
A population undergoes a split forced by some environmental changes, developes differences, and once the barriers cease to exist are not able ot interbreed anymore.
Hybrid zones are considered to be zones of secondary contact where the species have not separated completely. Example: fire-bellied toad (Bombina variegata - Bombina bombina) hybrid zones.
A widely distributed population responds to geographically variable selection in a way that we can recognize a cline. If selection against the wrong allele is strong enough and if those loci also contribute to reproductive isolation, the populations can differentiate into reproductively isolated species.
It is difficult to differentiate if hybrid zones are in the process of parapatric speciation or secondary contact.
A population splits into two species in place. One can imagine a insect that feeds on plants and at one point in time we have at a specific locus an allele A that prefers plant A and allele B that prefers plant B. Homozygotes AA and BB will be feed on different plants, and further assume that the heterozygotes AB are not well adapted and are less fit. If another locus (with alleles R and r) defines where to lay eggs and mate is also tied to the same host plants and these two loci are closely linked, reproductive isolation can occur.
When two populations are isolated from each other, this will affect all loci. Mutations will increase the differences between the two populations and by chance will also change the loci that are responsible for courtship/ mating etc. When the two populations come together again the loci for reproduction may have achieved enough differences that there are no succesful matings between the two groups.
Closely related species frequently differ by reaarangements such as fusions and fissions of chromosomes (Robertsonian translocations), but rarely involve translocations, which have a higher disadvantage. But the latter would make it much easier for us to see that we have two different species.
Peripheral populations can emerge as new species by rapid changes in the whole genome in very small isolated populations (Mayr (1954, 1963): Genetic revolution, Peripatric speciation). The selection pressure will be different on the different small populations, because the environment for that small localized population will be more homogenuous than for a larger more widespread population.
Carson (1975) proposed a similar scheme in which founders (e.g colonizers of an island) expand rapidly into empty space. Selection is relaxed and recombinants that had low fitness will become more prevalent. Upon crash of such a population genetic drift and selection determines which of the new recombinants persist.If two species who developed partially post-zygotic reproductive isolation meet gene flow and redcombination will eventually undo all the differentation. Selection for being different than the other species (Reproductive character displacement) would force the different species into prezygotic reproductive isolation.