Discovered by Edwin Howard Armstrong, which allowed for a high-quality reproduction of sound (referred to as high fidelity) to be played over the radio. It minimized interferences in audio like static, which was (and is) a major problem for AM frequencies.
FM is not just used in radio broadcasting, but was also used to transmit sound for early television.
Exact month and date of the discovery of FM unknown.
For more milestones in the development of radio, see:
Source: Brian Wilson, Media Technology and Society: A History from the Telegraph to the Internet (New York: Routledge, 2005), 79.
Image Source: “Edwin Howard Armstrong (1890-1954),” IEEE History Center, http://www.todaysengineer.org/2008/Dec/images/history-pic.jpg (accessed December 14, 2011).