One who conforms to a belief or set of beliefs held by someone else. For example, as many of the definitions on this page have noticed, there are adolescent social groups in Western society that many people feel the need to conform to, such as goths, punks and emos for example. However, some people may be forced to conform to a set of beliefs that they do not necessarily wish to follow; for example, in dictatorships like the Soviet Union and Nazi Germany in the 20th century, people were forced to conform to what the government believed, regardless of what they themselves thought.
The realm of conformism can be stretched, however. For example, one might argue that by being a non-conformist, you are conforming to non-conformism. However, this is bullshit and you should severely injure anyone who truly believes in that. There is no such thing as the non-conformist group in the same manner as there are punks and emos.
Of course, you could be a non-conformist by not conforming to the social norms as defined by adults, not adolescents, but nobody gives a shit about those anyway.
A conformist is what most of the people who have put entries onto urban dictionary are. You're not a non-conformist simply because you laugh at conformists. A non-conformist is a true rebel, fighting against whatever the prevalent social groups are in their society. In modern Western society, this essentially means wearing whatever you want, listening to whatever music you want, reading what you want and so on, not choosing a social group (such as the emos) and shoehorning yourself into that particular style. For example; ask an emo exactly why they like emo music or emo clothes and don't stop until you get a satisfactory answer. No, wait, don't, because you'll die long before you do (unless they really do like their clothes and music for real reasons, which is unlikely).
Someone who likes whatever music they actually like is a non-conformist. If someone likes music from any genre simply because they like that particular piece of music then they are a non-conformist in that particular area of life.