The strict sense of remix involves someone (usually legally) creating a new mix using the recorded elements of someone else’s song. Originally remixes were extended dance versions of song for clubs, and this is still probably the most common kind of remix.
Remixes are usually authorized by the artist or copyright holder of the original, and typically the source tracks are provided to the remixer for selective use and manipulation. Bootleg or white label remixes can be bought on vinyl but are not licensed. Today many people use the term remix to refer to remixes done without authorization or access to the original master tapes (or digital files).
See also: mashup (audio)