The minute a person writes down the lyrics or records the melody to a song, that person owns the copyright to his song. No one else may play this song in a public setting and/or make a profit off it without the permission of the copyright holder. Copyright for songs lasts for the lifetime of the copyright holder plus 70 years after his death. After this time, the copyrighted music enters the public domain, meaning anyone may record the song and profit from it without asking for permission.
While a songwriter owns the copyright to a song after writing the words or making a recording, verifying this is in court requires proof. To make a copyright official, songwriters must register their songs with the U.S. Copyright Office. By submitting an application and a fee, a songwriter can at least prove the date on which she registered her songs with the U.S. Copyright Office. The "poor man's copyright," mailing a copy of a song to oneself, therefore postmarking the song, does not hold up in court, so registering with the U.S. Copyright Office is a strong rule to follow in ensuring copyright.
Writing a song with a melody or lyrics very similar to a copyrighted song and making a profit off it is just as bad as attempting to profit off performing a cover version of a copyrighted song without permission. Copyright rules protect a songwriter from other writers' creating derivative works. This applies to lyrics and melody. For example, Bright Tunes Music sued former Beatle George Harrison for using the melody to The Chiffons' "He's So Fine" in his song "My Sweet Lord." This case proved that regardless of intent, creating a song too similar to a copyrighted song is illegal.