Though something that was around before it's dubbed 'start' the term "Straight Edge" derived from ( or has been said to be from) the song "Straight Edge" by band Minor Threat.  Coming to the surface in the late 70's and early 80's when most hardcore and or punk shows were drug and alcohol fueled ,  it began as the rebellion against such acts. Since then many well known acts have been part of the Straight Edge culture. Commonly found in heavier music setting, bands today who are Straight Edge can range from classical to pop to even country. 

Here is a playlist of some Straight Edge bands: