A songwriter is an individual who writes both the lyrics and music to a song. This tag can be used for people who do just that.

Although songwriters of the past commonly composed, arranged and played their own songs, more recently the pressure to produce popular hits has tended to distribute responsibility between a number of people. Popular culture songs may be written by group members, but are now often written by staff writers: songwriters directly employed by music publishers.

Someone who solely writes lyrics may be called a lyricist.
Someone who only writes music may be called a composer.

Someone who also sings the songs he/she has written and composed is generally known as a singer-songwriter.