Softcore punk rock.
"Guy Picciotto formed Rites of Spring in 1984, breaking free of hardcore's self-imposed boundaries in favor of melodic guitars, varied rhythms, and deeply personal, impassioned lyrics." Wikipedia
It is a label people give themselves for being apathetic and emotional to an extreme (usually miserable), but wallowing in it and not wanting it to change. They tend to feel they are misunderstood and that life is not fair to them in specific.
The casual Emo garb is faded thin\tight blue jeans or slightly oversized work pants accompanied by a secondhand T-shirt with an out of place logo, bad Heavy Metal tees work well too. Good shoes would be Converse Chuck Taylor low-tops or old skool low-top Vans. The more extreme clothes of choice would include horn-rimmed glasses, cardigan sweaters, dorky polo shirts, pants that are a bit too short and hair that is spiky or messy in the back with straight cut bangs or awkward sharp angles and are often glossed.
Compared to Goths:
The primary difference is that emos have normal or more or less standard fashion, personalities and interests. Goths take these to the extreme and dislike most people. Emos can have a pale complexion, but will not wear black lipstick, and will not have the Halloween look that goths have. Google images gives accurate portrait of goth.
Rites Of Spring
Thursday - Full Collapse
Saves The Day - Stay What You Are
Dashboard Confessional - The Places You Have Come To Fear The Most
Further Seems Forever - The Moon Is Down
Jimmy Eat World - Jimmy Eat World (Bleed American)
Sunny Day Real Estate - LP2
Texas Is The Reason - Do You Know Who You Are?