Emotive Hardcore is an offshoot from hardcore punk which formed in the late 80s and was played throughout the 90s. The term "emo" was coined to such bands like Rites of Spring and Embrace and other bands with a more melodic hardcore sound and expressive lyrics, although some later bands experimented with a softer indie alternative sound as well. You can also say it was a term to describe the "sound" of the DC area of the time. However, if you ask any of the members, they would highly disapprove of this term and most likely mock it.
Emo is not dying your hair black, piercing your lip, and it's certainly not cutting your wrist, that's just a trend. Emo is a music genre that started in the 90s and ended in the 90s.
-Rites of Spring
-Cap N' Jazz