A sub-genre of rock music closely related to punk.
Other sub-genre's similar or easily mistanken to be emo could be Post-Hardcore or Pop-Punk.
Emo (or "Emo Kids") also refers to people who listen to emo, (usually teenagers). The common stereotype with emo's are that they are deppressed, bisexual, fake, self loathing, dark, sensitive, and extremely associated with self harm. This is not always the case. Yes, some emo's are deppressed, but not all of them. Some are bisexual, others arent. etc, etc, etc... Those stereotypes can apply to other sub cultures, not just emo.
There is a large number of emo kids that arent like that. There are quite a lot of happy, confident emo's.
Emo bands: My Chemical Romance, Hawthorne Heights, Taking Back Sunday, Dashboard Confessional.