Natural selection explains how a population changes, but Darwin's title promised more: the origin of species, the splitting of one kind of life into many. That branching is what turns gradual change into the vast diversity of life.
What a species is
A useful working definition: a species is a group of organisms that can interbreed and produce fertile offspring, but cannot do so with members of other such groups. Lions and tigers are separate species; a population of finches on one island that freely interbreeds is one. Speciation, then, is the process by which a single interbreeding group divides into two that no longer mix.
Isolation and divergence
The usual recipe has two steps. First, a population is split so its parts stop interbreeding: most commonly by geography, when a mountain rises, a river shifts, or a few strays colonise a distant island. Once the halves no longer exchange genes, they evolve on their own. Different environments impose different selection, and chance changes accumulate independently, so the two groups diverge, drifting apart in form and behaviour.
The point of no return
Given enough time apart, the differences pile up until the two populations can no longer interbreed even if reunited: their mating signals, timing, or biology no longer match. At that moment one species has become two, and the split is permanent. Darwin's finches on the Galápagos are the classic illustration: a single ancestral finch, spread across islands and adapting to different foods, radiated into many species with distinct beaks, each suited to its niche.
The tree of life
Repeat this branching over billions of years and you get the tree of life: a single trunk splitting again and again into limbs, branches, and twigs, most of them dead ends, a few flowering into the millions of species alive today. Every fork is an ancient speciation event; every living species is a twig on that vast, branching bush. This is why common ancestry and the diversity of life are two sides of one coin: the same branching process that separates species is what unites them all in a single genealogy. The final lesson turns to the evidence that this tree is real.