Just to clear things up a little bit, I'll do a quick definition of terms so nobody can mix up emo, screamo, hardcore and grindcore/old school.
Emo : A music gender influenced majorly by emotions. The singer normaly seems to cry desperation and the band is mediocre; easy guitar riffs and bad drum playing are required
Bands related : My chemical romance, Taking back sunday, etc.
Screamo : Emo took to another level. You got double pedal added to the drum, some guitar solos (not outstanding though) and the singer mix up scream and singing.
Bands related : Underoath, Alexisonfire, Funeral for a friend, etc.
Hardcore : This, compared to those below, is completely different. Instead of having "girly tight pants wearing" musicians, you finally enter in a world where guitar, bass, drum players and singers looks normal. You start getting into music without boundaries, everyting is possible and you can get as experimental as you want. 2 steps, breakdown and moshing kicks in to give you a great musical experience.
Bands related : Old Atreyu, Old As I Lay Dying, Haste the Day, Still Remains, Between the Buried and me, etc.
Grindcore/Old school : If you get here, you can never go back. This is the most violent style of music you might ever heard, a combination of hard punk, death metal and hardcore. Almost everybody stops at hardcore and will probably won't discover how precious this last style is because it is impossible to get records of those bands without going to their shows. Hard to describe, you need to hear it to make the difference between Hardcore and Grindcore.
Bands related : Ion Dissonance, The Tony Danza Tap Dance Extravaganza, Into the Moat, Number 12 looks like you, Despised Icon, Sworn enemy, etc.
Some of this is opinion based, but I did my best to help everybody know what they are listening to.
<3 Luigitoilet 'cause he likes Between :D