Why do vertebrates share a common ancestor?

Vertebrates is the name of a taxon, a clade in Biology. The most important defining characteristic of this group is that they all have vertebrae.

Within this group are many subgroups, which have slightly different characteristics, but all share the defining characteristics of the larger group (the vertebrates).

The group vertebrates itself is a subgroup of a larger group. Together with a couple of other groups, it is part of the group Chordata. The most important defining characteristic of this group is that the animals in it all have a central nerve chord along the length of their bodies. Vertebrates and all its sister-groups share this trait, and all the other traits of the larger group, but differ in others (eg. not all chordates have vertebrates).

It is this pattern of nested hierarchies, sets within sets within sets, each set sharing the defining characteristics of the superset, but differing in others, that made naturalists conclude that the diversity of life is the product of a process of continuous divergence: evolution. In this model, a superset (ancestral form) produces lineages (subsets) with diverging characteristics. Each of the diverging lineages keeps many of the characteristics of the ancestor, but becomes increasingly different from the ancestral form and its sibling lineages in other aspects. It is this pattern that tells us that all vertebrates have a common ancestor that was part of the clade of Chordata.