The neckbeard is a feature of the slovenly obese, that stereotypical virginal denizen of teh intarwebs who lives in his parents' basement at age 25+ and despite his college education and vast computer skills, works a menial, low paying, low stress job, perhaps as night manager of a grocery store or phone tech support.
The term neckbeard brings up a host of unflattering imagery. Most directly, it refers to the coarse, greasy stubble accumulating around the neck resulting from a lack of bothering to shave, despite not having much facial hair on the actual face to speak of. Occasionally this is done on purpose to help hide a double-chin. It doesn't work, by the way, it just looks like a hairy double-chin.
But one step removed from this is the plethora of insinuations that follow from being the sort of person who would have a neckbeard in the first place. The bearer of a neckbeard is not overly concerned with his appearance, or he would shave. This implies that he also doesn't bother to bathe regularly, and has no reason to care about his appearance because of the limited direct contact he has with other people. Again, a following implication can be drawn that he has few, if any, friends in meatspace and therefore certainly no girlfriend.
For some reason I still do not fully understand, it is also to be assumed the person has several t-shirts featuring wolves.
The general process that is thought to produce neckbeards runs like this: A person is a nerd and a loser, and has few friends. Upon graduating from high school or college, low self-esteem brought on by near-constant torment in school causes the person to withdraw, eschewing contact with other people. Not being in contact with others, there is little reason to maintain hygiene, and a sedentary lifestyle (television, video games, and internet forums being his primary pastimes) coupled with unhealthy convenience food causes weight gain. Shaving drops to a relatively low position on the priority list, but lack of testosterone (nerds are not manly enough for real beards) results in patchy, wispy facial hair which may even go unnoticed. The inevitable end result of all this being the neckbeard.
Neckbeard has come to be a popular insult on various internet forums and discussion boards. The easiest assumption to make about someone across the internet is that the target spends all of his copious free time using computers, and therefore is a friendless sad sack as described above.
But of course, as with all such things, there are individuals who have embraced the term neckbeard, either as a source of pride in oppression, or out of a sense of irony. Intentionally shaving one's facial hair to leave a fuzzy neck is not unheard of, and the results can be somewhat odd depending on the quality and quantity of the individual's beard. Proponents are quick to point out that Henry David Thoreau sported a neckbeard, going so far as to include this fact on Thoreau's Wikipedia article. Still, I don't expect it to go mainstream any time soon.
artman2003 adds that Canadian actor Seth Rogen is famous for playing quintessential neckbearded characters, such as Ben Stone in Knocked Up (2007).