Search through the liner notes of the album (if you own a physical copy) for songwriting information. Musicians usually list the songwriting credits somewhere on these inserts or the packaging of their music.
Access the song's information if you have a digital copy of it. The easiest way to access a song's info is to use your preferred media player. Click the "Get Info" button on iTunes to get a song's info. Or, left click the name of a song in Windows Media Player, click the "Properties" tab, and then click the "Content" tab at the top of the pop-up window to access the song's info. The songwriters are generally listed under the "composer" section of the info page.
Search the Internet to help you identify songwriters. The American Society of Composers, Authors and Publishers (ASCAP) website and the BMI Repertoire website are easy-to-use databases that can tell you who wrote just about any song.