Fags of 4chan
The suffix -fag has probably started as a bad attempt at an insult somewhere back in the dark days of history. Soon it became so widespread and applicable to any form of belief, subculture, activity and so on, that it became obsolete. Today, the suffix -fag is just a slangism that carries no fixed positive or negative connotation by itself, and is used to describe someone simply as belonging to a certain group of people. As far as meaning is concerned, it could be replaced by something like -person (Newfag -> New person). Whether a certain *-fag is used as an insult or not is instead derived from a) the context of the discussion, especially the person using the word and b) the part preceding the suffix and not the suffix itself. It can even be used as a sign of affection.
It has been noted that due to the suffix of -fag on 4chan can be as demeaning or endearing as the word "child" is to normal people. With an implication of man-sex.
For example, a person from Europe might refer to himself as a Eurofag, with a neutral tone, simply indicating that he hails from Europe. "Yeah, I'm an Eurofag, so I can't watch American football on my TV."
A different (probably non-European) person might use it as an insult. In this case it is the euro- part that serves as the carrier of all known negative stereotypes and connotations about Europeans, from the insulting person's cultural viewpoint "Shut up Eurofag, go suck some Muslim cock."
A third usage is almost the same as the first example, but with a more positive intonation. "Eurofags make the best music."
Namefag
Someone who attributes a name to their posts on 4chan. Namefagging is generally considered a terrible thing, though /tg/ is more relaxed than some other boards. Basically, don't do it unless you need to be distinguished from other Anonymous.
Tripfag
Another way to differentiate from Anonymous. Tripcodes are like little keys, to show people that the other guy posting as BobsUrUnkle isn't really you. Much like namefagging, you should avoid this unless you are either a person of note outside of 4chan (such as the ghost of Gary Gygax posting on /tg/), or you use it to establish identity for a certain project (such as Weaver when running RubyQuest). No other purpose is acceptable.
Drawfag
A person who doodles images of varying relevance, sometimes with a surprising degree of skill. See the main article here.
Newfag
A poster who has allegedly arrived at the current board or topic later than the person using the term. See also: newfag summer, a mostly imaginary menace people blame on dumb users.
Touhoufag
Touhoufags are fans of the Touhou video game series, which you can ask /jp/ about. Touhoufag is a specific personality on /tg/ who handles 4e.
(whatever)fag
Any one person related in any way, shape or form to the (whatever) in the title. You can put anything there, seriously.
Writefag
Any person on a message board who likes creating stories.
Fag
Homosexual male. Sometimes used as a generic insult, though usually with little effect. Can be referred to as a Fagfag. Also a slang term for a cigarette.
++Faggot is also an archaic term for a bundle of sticks (Which is most likely where the shortened 'fag' as colloquial language for a cigarette originated) or a bassoon++ ++As well as for meatballs. Not so archaic, though++