Allopatric speciation occurs when geographic isolation prevents gene flow, allowing populations to diverge via drift and local selection. Peripatric speciation is founder-effect-driven rapid divergence. Parapatric speciation occurs with ongoing gene flow. Sympatric speciation occurs without geographic isolation, often via polyploidy (plants) or disruptive selection.
You already know that speciation requires reproductive isolation — the buildup of barriers that prevent two groups from interbreeding. The modes of speciation are defined by the geographic context in which that isolation develops, and understanding this context helps you predict how fast, how common, and how genetically distinct the resulting species will be.
Allopatric speciation is the classic and most widely documented mode. A physical barrier — a rising mountain range, a river changing course, glaciers advancing — splits one population into two geographically separated groups. With gene flow completely severed, each population accumulates genetic changes independently through natural selection acting on local conditions and through genetic drift. Over thousands or millions of generations, the two populations diverge enough that they can no longer interbreed even if the barrier disappears. The finches of the Galápagos Islands are a textbook example: ancestral birds colonized different islands, and isolation on each island drove divergence in beak shape, body size, and mating signals.
Peripatric speciation is a special case of allopatric speciation involving a small founder population that becomes isolated at the edge of the parent species' range. Because the founder group is tiny, genetic drift is exceptionally strong — rare alleles can become common by chance alone, and the population can shift rapidly to a new genetic and phenotypic state. This is sometimes called the founder effect. Island colonizations often fit this model: a handful of individuals blown to a remote island carry only a fraction of the parent population's genetic variation, and the resulting population may diverge quickly. The key distinction from standard allopatric speciation is the asymmetry in population size and the outsized role of drift.
Parapatric speciation occurs when populations are adjacent and exchange some migrants, but a strong selection gradient across the landscape overwhelms the homogenizing effect of gene flow. Imagine a grass species growing across a boundary between normal soil and soil contaminated with heavy metals from a mine. Plants on contaminated soil face intense selection for metal tolerance, and if that selection is strong enough, the two adjacent populations can diverge despite some cross-pollination at the boundary. A hybrid zone — a narrow strip where the two forms interbreed — may persist indefinitely, or the two forms may eventually become fully reproductively isolated. Parapatric speciation is harder to demonstrate than allopatric speciation because you must rule out the possibility that the populations were once fully isolated and only recently came back into contact.
Sympatric speciation — divergence without any geographic separation — is the most controversial mode because gene flow within a single population should constantly remix alleles. Yet it demonstrably occurs, especially in plants through polyploidy: a mutation doubles the chromosome number, instantly creating an individual that can breed with other polyploids but not with the parent species. In animals, sympatric speciation is rarer but has been documented in cases of strong disruptive selection, where extreme phenotypes have higher fitness than intermediates, combined with assortative mating. The cichlid fishes of African crater lakes, where dozens of species have arisen within a single small lake, are among the most compelling animal examples. The spectrum from allopatric to sympatric is really a continuum of gene flow levels during divergence, and many real speciation events likely fall somewhere in between the textbook categories.