Life is unfair; kill yourself or get over it.—Black Box Recorder, "Child Psychology"
I stole some pins from the noticeboard and pressed them into my hand and they spelled "why?"Because there weren't enough pins for "oblivion"
Why did they spell "why?"
—Bill Bailey, "The Song Written from the Perspective of a Young Man Who Works at Starbucks and Self-Harms"
Teenagers who are whiny, self-important, and generally trying to work their way to self-identification.The character type based off the Emo persona.
The things that differentiate and identify Emo Teens are that they dress in black, listen to strange music (specifically, the "emo" genre, whence the term comes) and generally think that they are erudite (or, at least, surprisingly coherent). The slightly better-kept ones are usually political. If the character is female, she will tend to be extremely self-conscious about her appearance, but implausibly attractive all the same.
Since Emo Teens cultivate an out-of-the-mainstream image, sometimes writers get lazy and use them as a stand-in for any variety of geekishness that isn't considered "too nerdy".
Most shows with an Emo Teen do it semi-sympathetically, where the character is at worst whiny and judgmental, but fundamentally good. However, if any character is going to suffer from an overload of Wangst, it's them.
Note that in Real Life most definitions of "emo" require that you act like your life sucks when it doesn't to qualify, but since There Are No Therapists, no matter how dark a Backstory a fictional character has, if he acts even remotely unhappy with his lot in life, he is "emo". Internet fanboards don't care if you spent the first decade of your life in a cupboard under the stairs only to discover your magical new world comes with a mortal enemy, your dad scarred your face with fire and kicked you out for valuing human life above military victory, or you came home from school one day to find that your big brother killed your parents. Suck it up, emo boy.