When you talk about copyrights in the music industry, at least two different copyrights exist.  There is a copyright in the lyrics and/or music of the song [the musical composition], and a separate copyright in the recorded version(s) of the song [the sound recording(s)].  If you record or re-record a song (your own or a cover of someone else’s), the copyright in the lyrics and music would remain the same, but there would be a new copyright in each new sound recording. 

Copyright in Music and Lyrics

Under the U.S. Copyright Act, everyone that contributes to the songwriting has a claim to the copyright.  Regardless of how much or how little you contribute, the Copyright Act considers all contributors to the same song to be joint authors (a.k.a. co-authors).  Each joint author owns an indivisible share to the entire copyright, unless there is a written agreement stating otherwise between all of the contributors (or any non-contributors).  Therefore, each co-author can do what he wants with the copyright as long as he pays the other co-author(s) an equal share of the proceeds.  Copyright ownership of a song can be very profitable, as songwriters receive money from publishing.  (Sometimes, songwriters assign their copyrights to music publishers to exploit the songs and administer the rights.)

Copyright in Sound Recordings

Technically everyone that contributes to the recording has a claim to the sound recording.  However, usually whomever pays for the recording (usually a record label) owns this copyright through a contract.  The songwriter will own the copyright in the lyrics and music, but the record label will own the copyright in the sound recording.  If the artist pays for the studio time, then the musicians who play on the recording will jointly own the copyright to the sound recording unless this is a written contract to the contrary.  

If the same person or group of people own the copyright in one or more musical compositions as well as the resulting sound recordings, then both copyrights can be protected on one copyright application.  If the owners are different, then separate copyright applications will need to be filed for the musical compositions and sound recordings.