Apomorphy is a term used to describe any "interesting" hereditary character of a species. It is used specifically in the context of evolution and phylogeny, in the sense that the apomorphies of a species are those novel adaptations that define the species. Of course, what constitutes "interesting" or "novel"? The more unusual or complex the better. Feathers, deformed flower parts, and specialized enzymes are all good candidates; leaf shape, hair color, and feet are probably not so useful.
If shared with another species, an apomorphy can be used as evidence of shared ancestry. In particular, a synapomorphy of a group is one shared only among those species and no others. Its uniqueness makes it a highly useful character in proving the group to be monophyletic. (That's everybody's goal in life, right?) Feathers, for example, occur only among birds and nowhere else, proclaiming fearlessly to the world that all feathered creatures are cousins. Although less dramatic, the peculiar microscopic variations in shape of the stigma and anther in the family Asteraceae help the patient botanist just as reliably to separate the nearly 25000 species of sunflowers into twenty-odd tribes.
If, on the other hand, the character also occurs in species outside the group in question, then it is not so useful. These other species are often distantly related to the group by an older common ancestor, so this trait can be considered ancestral or primitive, and is then called I didn't make these words up, folks a plesiomorphy. [Need example.]