Me? I like the Federal Agencies Digitization Guidelines Initiative's definition for derivatives:
Often called service, access, delivery, viewing, or output files, derivative files are by their nature secondary items, generally not considered to be permanent parts of an archival collection. To produce derivative files, organizations use the archival master file or the production master file as a data source and produce one or more derivatives, each optimized for a particular use. Typical uses (each of which may require a different optimization) include the provision of end-user access; high quality reproduction; or the creation of textual representations via OCR or voice recognition. In many cases, the derivatives intended to serve end-user access employ lossy compression, e.g., JPEG-formatted images, MP3-formatted sound recordings, or RealMedia-formatted video streams. The formats selected for derivative files may become obsolete in a relatively short time.
Or, to be a little shorter and less precise, derivatives are copies of files in different formats that serve both archival and display purposes. In Islandora they are usually created when the asset is ingested into the repository, but they serve several purposes. Derivitives are a pretty standard approach when creating archival packages of content, and disseminating them to the public (for example, derivatives are usually a part of AIP, SIP, and DIP packages).*