A remix is a type of derivative work in which the the original song is transformed in some way to create a variation of the original. These transformations lead to a new creative work that is noticeably different from the original. The changes may be subtle, but someone with a trained ear must be able to tell the remix apart from the original. If you can't tell the them apart then you haven't done anything meaningful to the original song.
According to wikipedia, "A remix is an alternative version of a song, different from the original version. A remixer uses audio mixing to compose an alternate master recording of a song, adding or subtracting elements, or simply changing the equalization, dynamics, pitch, tempo, playing time, or almost any other aspect of the various musical components. Some remixes involve substantial changes to the arrangement of a recorded work, but many are subtle, such as creating a "vocal up" version of an album cut that emphasizes the lead singer's voice. A song may be remixed to give a song that was not popular a second chance at radio and club play, or to alter a song to suit a specific music genre or radio format. Remixes should not be confused with edits, which usually involve shortening a final stereo master for marketing purposes."
Remixes as we know them started in Jamaica with dub music. This was soon followed by disco and Hip Hop remixes. The remix was a form created for the dance floor, but the story begins decades before that with experimental composers and engineers like Pierre Schaffer and Pierre Henry. Today remixes are not just for the dance floor. You could say they have gone full circle. In terms of the remixin project, any type of remix along this spectrum of derivative works is acceptable. We encourage diversity!