"Derived" in this case means that its ancestor again is lacking it — so it is a derived (new) character-state, or apomorphy, originating in their last common ancestor. Consider species A and B, their common ancestor C, and C's ancestor D. If A and B have trait X, and C did as well, but D did not, then X is a synapomorphy: a shared derived (new) character-state, or apomorphy, originating in A and B's last common ancestor, C. True synapomorphies usually uniquely characterise a given set of terminal groups, but this is not essential to the concept. In cladistics, synapomorphies are used to establish phylogenies. As such they are empirical data which can support a certain hypothesis that terminal groups form a clade (monophyletic group) together to the exclusion of certain other groups, whereas character-states that are shared, but also shared by other terminal groups descending from an earlier common ancestor, cannot be used to exclude these other groups. The latter character-states can consist of symplesiomorphies ("primitive" character-states having originated in the earlier common ancestor) or homoplasies (superficially similar but independently evolved derived character-states).
The synapomorphy is thus opposed to both the symplesiomorphy and the homoplasy. Rather uncontroversial examples of these three are:
The key problem is to identify the polarity of the transformation series to which several character-states belong, i.e. to tell which character-state is apomorphic and which plesiomorphic. To polarise the transformation series in earlier cladistics various criteria were used; however in the recent two decades the pattern criteria based on outgroup comparison dominate the field.
The concepts of apomorphy and plesiomorphy are relative to a certain level of generality. What counts as an apomorphy at one level of generality may well be a plesiomorphy at the other. E.g., for rats and apes, the presence of five toes on their legs is a symplesiomorphy, but for Tetrapoda in general it might be a synapomorphy.
It is not essential to a synapomorphy that all members of a clade possess it; even if some would have secondarily lost the trait it could still be a synapomorphy of the clade as a whole. A character state that is a synapomorphy for a clade, but for lineages in this clade is a plesiomorphy that is altered in some lineages, is called underlying synapomorphy. If no stem group taxa are known, it is sometimes difficult to decide which character state is the underlying synapomorphy and which the autapomorphy that overlies it.
Clades are not defined by synapomorphies as such, though it is possible to define them by apomorphies in general.
The word synapomorphy is derived from the Greek words σύν, syn = with, in company with, together with, ἀπό, apo = away from and μορφή, morphe = shape.