A fan is someone who is an enthusiast of something. The term is short for 'fanatic.' The standard plural is 'fans,' though occasionally the plural 'fen' is used. A fan is usually distinguished from a person who merely likes something by extreme levels of passion for the subject beyond mere approval or enjoyment. Fans are often driven to speculate, argue about, defend, and generally vocalize about the object of their fandom, creating stories, art, or just discussion. Fans are often also characterized by an in-depth knowledge of their fandom, since their passion drives them to spend a lot of time re-reading or -watching the work, and also thinking about it and discussing it with other fans.
Just like there are effective people and ineffective people, experienced people and inexperienced people, and knowledgeable people and ignorant people... all of these exist among fans as well. No matter how annoying it can be, young, rabid, senseless, or stupid fans are still fans no matter how much others wish they were not.
It is important that no fan be turned away from a fandom if they really and truly wish to improve their work. Young fans often grow out of their awkwardness or inexperience, and anybody can make an effort to be less annoying.
Also, there is no such thing as a 'true fan'. Often internet arguments over fandom will claim a party is obviously a lesser or less pure fan than the accuser. This is pure nonsense: the amount one knows, the number of episode viewings, or even a greater quality of fanfiction does not qualify anybody as a superior fan. All fans are equal, for they're all passionate about something in their fandom, and that cannot be measured.
However, their tolerability and the products they produce are not equal. Selfish fans who care only about one element of their fandom at the expense of all others spoil everybody else's fun and are right to be disliked and have their work satirized. Also, while one's depth of knowledge may not qualify anyone for different degrees of fandom, it is expected that someone who claims to have great passion for a work at least appear to be familiar with the basics. If a person hasn't actually seen any episodes, or doesn't respect what they're about, it's sort of impossible to be a fan of them. Claiming to be a fan in the face of blatantly not knowing or even respecting anything about the canon - i.e., not being a fan - makes said person's work an acceptable target.