There are many different types of licenses to grant certain permissions depending on the intended use for the musical piece. See a short summery of the most common licenses below.
Master Use License
(also: Master License, Original Master Recording License)
A master use license is between the master owner (usually the label or the recording studio) and the licensee.
It grants permission to use a musical piece already recorded in a visual project. A synch license is also needed along with a master use license to grant full permission to use a recording. Like synch licenses, master use licenses are used in television shows, movies, commercials and any other video that requires music.
(also: Synch License, Sync License)
A synch license is between the copyright owner(s) of the underlying composition (composer/author and publisher) and the licensee.
It grants permission to use a musical piece already recorded in a visual project.
Synch licensing fees depend on many factors like the fame of the song, the fame of the performer, the way the song is used in the production (featured, background, source, etc.), the length used, the media the production will be presented in, etc.
It is an agreement with the copyright holder, the publisher, the songwriter or the appointed representative (in the US most often the Harry Fox Agency, and in Europe most often the PRO) that allows the holder to reproduce the sound recording. A mechanical license is a (often required) license that grants certain limited permissions to work with, study, improve upon, reinterpret, re-record (etc.) something that is neither a free/open source item nor in the public domain.
A mechanical license gives the holder permission to create copies of a recorded song which they did not write and/or do not have copyright over.
A performance license is between the PRO and the licensee.
A performance license is obtained by paying a (mostly annual) fee to the local PRO, supported by a cue-sheet listing the publicly performed songs. This is needed for all public events where music is used or played. This includes online streaming, broadcasting, etc. and in some countries digital downloads too. It's the organizer's (or website operator's) responsibility to purchase the license and file the cue-sheet.
A print license (between composer/author and licensee) grants a party permission to copy or print musical lyrics or any type of written music for self use or reproduction. A print license is needed for each song being copied or printed.