Nothing pisses me off more than seeing bands like the Used or Atreyu called "emo." True emo is derived from hardcore, and screamo draws a lot of influence from, of all things, grind.

NOT EMO:
Atreyu (metalcore)
My Chemical Romance (pop-punk)
the Used (pop-punk)
Fall Out Boy (pop-punk)
Eighteen Visions (used to be chaotic metalcore, now hard rock)
Avenged Sevenfold (metalcore, then NWOAHM, now power metal[?])
Thrice (melodic hardcore)

True Emocore:
Saetia
City of Caterpillar
Rites of Spring

True Screamo:
Circle Takes the Square
Kaospilot
Pg.99
I Would Set Myself on Fire for You

There is also a genre called "emo-violence" which mixes the emotion and metaphor of screamo with the music of power-violence (i.e., Spazz, Charles Bronson).

Emo-Violence:
combatwoundedveteran
Usurp Synapse

There, now stop calling shitty pop-punk "emo"