Peripatric speciation is a special case of allopatric speciation in which a small founding population colonizes a new area. Genetic drift in the small population rapidly changes allele frequencies and may quickly establish reproductive isolation even from the ancestral population. Island colonization exemplifies peripatric speciation.
You already understand allopatric speciation — populations separated by a geographic barrier diverge independently until they can no longer interbreed. Peripatric speciation is a specific version of this process where the key ingredient is not just geographic isolation but extreme smallness of one of the populations. Imagine a continent with a large, stable population of birds. A storm blows a handful of individuals to a remote island. Those few birds carry only a fraction of the genetic variation present in the mainland population — this sampling effect is the founder effect, which you encountered in your study of genetic drift.
The founder effect sets the stage, but drift does the heavy lifting. In a population of millions, rare alleles stay rare and common alleles stay common because the law of large numbers keeps frequencies stable. But in a founding population of, say, 20 individuals, random chance dominates. An allele that was present at 5% on the mainland might end up at 25% or even 50% on the island purely by chance. Some mainland alleles will be missing entirely. Over just a few generations, drift can push the island population's genetic composition far from the ancestral state — changes that would take thousands of generations in the large mainland population.
This rapid genetic reshuffling can produce reproductive isolation surprisingly quickly. If drift alters alleles involved in mate recognition, gamete compatibility, or developmental timing, hybrids between island and mainland individuals may be inviable or infertile even if the two populations come back into contact. The small population may also face novel selective pressures in its new environment — different predators, food sources, or climate — accelerating divergence further. The combination of drift in small populations and selection in new environments creates a one-two punch that can drive speciation much faster than the gradual divergence typical of large allopatric populations.
The classic examples come from oceanic islands. The Hawaiian honeycreepers radiated from a small number of founding finch-like ancestors into dozens of species with dramatically different beak shapes, diets, and behaviors. The Galápagos finches that inspired Darwin followed a similar pattern. In each case, small founding populations on isolated islands diverged rapidly from their mainland relatives and from each other. Peripatric speciation helps explain why islands and isolated habitats are often hotspots of endemic species — geographic isolation plus small population size is a potent recipe for rapid evolutionary change.