In biology, an apomorphy is a characteristic of an organism that is not shared by a larger group it belongs to, nor by the common ancestor of the larger group. So it is an innovation, evolved after the organism's line diverged from the group ancestor.
A synapomorphy is an apomorphy that is shared by more than one organism. It indicates that the organisms that share it are more closely related, and they had a common ancestor which innovated the apomorphy.
Synapomorphies are the features used in strict taxonomy to set up clades, or groups with common ancestry.
For example, birds are generally distinguished by feathers and flight, and (probably) derive from dinosaurs which had neither. The placenta is the innovation that marks placental mammals out from other mammals. These features are synapomorphic to those groups.
The notion of synapomorphy is opposed to homoplasy, which is a feature in common because of convergent evolution. Birds, bats, and pterosaurs share adaptations for flight, but this is because of their lifestyle. They are three independent innovations of the same function. Flight is a synapomorphy for bats compared to other mammals, but is not a synapomorphy between birds and bats.
It must also be distinguished from symplesiomorphy. A plesiomorphy is a primitive condition of a whole group, that is one possessed by the group's ancestor. A symplesiomorphy is a shared plesiomorphy. It does not indicate greater relatedness. Whatever the relationship between branches A, B, C, and D of a clade, if C innovates an apomorphy, then the lack of that counts as a symplesiomorphy of A, B, and D.
An example of this is lampreys and hagfish, both jawless fish, traditionally grouped as Agnatha (Greek, = jawless) because of their similarity. The vast majority of fish have jaws. But jaws are an innovation, and jawlessness is probably just a symplesiomorphy of the common ancestor of all fish. Lampreys and hagfish might be only very distantly related to each other.
The terms are relative to a clade. As far as birds as a whole go, flight is a synapomorphy distinguishing them from relatives outside. But for the birds that have lost the power of flight, such as the ostrich family, flight is the symplesiomorphy, and their own ground-running lifestyle is the synapomorphy.
From the Greek elements morph = shape, plesio = near to, apo = away from, and syn = together with.
Colin Tudge, The Variety of Life, Oxford, 2000