In studio language, the master is the finished version of a recording from which all copies, downloads and streams derive. In rights language, the master is the sound recording itself as a protected object, distinct from the composition it contains. One composition can exist in many masters: a 1950s recording of a jazz standard and a new lo-fi version of the same tune are separate recordings with separate owners.
Master rights usually belong to whoever made and financed the recording, typically a label or a self-releasing artist. In the United States a sound recording is a copyright work of its own; in the EU, phonogram producers and performers hold related rights that last 70 years from publication, a term extended by Directive 2011/77/EU. Using a recording, for example by sampling it, requires permission from the master owner in addition to any permission needed for the composition.
Who owns the masters is a central question in recording contracts, so anyone reviewing a specific deal is best advised by a music lawyer.