Most people have a horribly limited idea of what emo is, simply because the most important records in the development of emo were largely released on on vinyl, in small numbers, and with limited distribution. These were however very influential, so nowadays you have the situation that a lot of kids listen to third- and fourth-generation emo styles without even knowing it. I hope to expose such people to a wealth of great preceding music that's getting easier to find all the time...
bands that are emo:

Taking Back Sunday
My Chemical Romance