...do you want to know what they were originally, or what they are now? I'll give you a brief sketch of both.
ORIGINALLY, Goth was an offshoot from Punk. In the beginning, as I'm sure you know, a lot of punks were kids who felt that mainstream society had no place for them. Well, the original Goth kids felt that even the punk scene had no place for them, and so they went elsewhere.
Originally, Goths were people who had been ostracized for being different. A surprising number of these people were depressive, and many others held onto ideas and concepts that were considered to be unacceptable (a few examples being the viability of socialism/fascism as political systems, an intellectual interest in the occult/ancient religions, or a distinctly un-cool appreciation for classic literature, film, and music). So rather than living poor and lonely, they grouped together and formed their own group. The black clothing and makeup has been given many meanings by outsiders, but the real reason for it was simply as a sort of uniform, so that they could identify each other - remember that in the beginning, the only difference between punk and goth fashion was that goth fashion was black.
It's worth noting that the vast majority of Goths were never particularly interested in nonconformity, violence, or really very much apart from living their lives as best they knew how and in the company of others who were like-minded.
NOW, there might be one or two 'real' Goths for every hundred kids you see in black eyeliner. The media and marketing gurus have gotten ahold of it, turned it into product, and mass-marketed it to the youth, the overwhelming majority of whom have never experienced ostracism, mental illness, or the genuine attraction of concepts deemed to be socially unacceptable. While the kids these days think they are Goths, I can tell you from personal experience that that which made the scene special, that which truly set the Goths apart, has been lost, diluted, or forgotten, and now it is simply a fashion trend that refuses to die out, and that gets sillier every year.
It may also be noted that a trend called 'cyber-goth' is apparently all the rage in the 'underground' these days. I have found that this 'movement' is essentially a continuation of the degradation of the original, with fashion getting sillier and sillier, and the music and art becoming more and more trivial with each passing day. Cybergoths can be identified by their wearing of bright colours, and can be distinguished from ravers by their wearing of store-bought hair and non-functional gas masks. (You'll know it when you see it.) Their aesthetic, both visually and musically, seems to stem from the idea that if one crappy synthesizer or cartoon android is good, then obviously a hundred must be a hundred times better.
Goth kids have been around more or less since the beginning in one form or another - I'd say since the late 70's in their earliest iterations. It has been periodically dying out and being reborn throughout this time, but in my personal opinion, the subculture proper died out for good in the early 2000's, and continues on as simply a 'scene' fostered by a few artists who refuse to give up, and a couple of big companies that refuse to stop beating the dead horse as long as there's some coin left inside.
How can you spot a Goth? You can't. They're gone, for all intents and purposes, and those that are still around really ought to be gone.
Classic Goth artists: Bauhaus, Sisters of Mercy, Joy Division, Siouxie and the Banshees. Modern Goth artists: older work by The 69 Eyes, also older work by VNV Nation (which can also be considered industrial), Johnny Hollow, Redemption.
As per Emo, ORIGINALLY it was another offshoot from Punk, populated by Punks who felt that the punk ethos of the time was far too narrowly focused on politics and what have you, and neglected to recognize the emotional state of the group. As a result they began making what was essentially punk, but with an emotional, rather than political, focus.
All fine and dandy, but the emo subculture proper didn't last very long (likely because they were regarded as being less than tough, and that doesn't get you far when you're hanging out with proper, old-school punks), and it faded into obscurity from the early 90's (at the latest) onwards.
For some reason it popped back up in the late 90's/early 2000's, but oddly, the new emos seemed to be highly concerned with dressing androgynously and talking at great length about their feelings, most of which consisted of being sad for no good reason. While this version enjoyed explosive popularity with the youth, it lay at odds with the original emo subculture, and, in my opinion, failed on a basic level to connect with what made the emo scene worthwhile in the first place.
There are still plenty of emo's around, though the fad is in the process of dying out. You can spot them by virtue of their boys looking like girls and vice versa. Also, they can be mistaken for Goths by the uninitiated, but it should be noted that no proper Goth would be caught dead in jeans unless they are deliberately attempting to provoke a reaction from their fellows, and sneakers are considered unacceptable footwear by Goths, who prefer either large, heavy boots, or impractically high heels.
Having never taken more than a passing intellectual interest in the emo phenomenon, I cannot with any reliability mention any significant emo artists.
It is also interesting to note that while both groups have been identified as having links to self-harming behaviour in the early to mid 2000's, I was in high school at the time and can vouch for the fact that self-harm was apparently some kind of trend or fad that swept the youth at that time and was not isolated to any particular group or groups.
