A plebeian is something no /mu/tant should ever be, but is regardless. Every one on /mu/, nay, 4chan fears being such a thing, but being is, in itself, inescapable.
Plebeian comes from the Latin word for a commoner. Essentially, a plebeian (pleb for short) is 4chan speak for being an idiot/wanker/dumbass/n00b and is the total opposite of a Patrician. It's taken in high offence to /mu/tants because, let's face it, we're all massive fucking hipsters.
One can identify a plebeian through their total lack of true patrician music in their last.fm chart, thinking that In the Aeroplane Over the Sea, and by extension, Neutral Milk Hotel is shit, and many other things.
Really though, the truth is, we are all plebeians because the term is as subjective as musical tastes. If someone ever tries to use it as an insult, they are the true plebeian, because they have not yet realised this fact.