What is a Derivative Work, and Why should I Care?

Marcel Duchamp’s derivative work of Leonardo’s Mona Lisa
On occasion I am asked about the extent to which a new work can incorporate elements of a pre-existing work without infringing the pre-existing work’s copyright. To answer such a question, one must understand derivative works.
17 U.S.C. Section 101 says (emphasis added):
A “derivative work” is a work based upon one or more preexisting works, such as a translation, musical arrangement, dramatization, fictionalization, motion picture version, sound recording, art reproduction, abridgment, condensation, or any other form in which a work may be recast, transformed, or adapted. A work consisting of editorial revisions, annotations, elaborations, or other modifications which, as a whole, represent an original work of authorship, is a “derivative work”.