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 preceeding the suffix and not the suffix itself. It can even be used as a sign of affection.
For example, a person from europe might refer to himself as an 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 need to keep yourself properly distinguished from others over a longer period of time.
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 can be discerned by two distinct features: 1) they supply a Touhou image with practicaly every post they make, usually making the image related through some weird logic and 2) on /tg/ they are usually min-maxers, optimizers and powergamers, searching for loopholes and exploits in RAW rules (although make it a point to severely discourage them), especially in D&D
(whatever)fag
Any one person related in any way, shape or form to the (whatever) in the title. You can put anything there, seriously.
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, what the connection is there we don't know.
++faggot is an archaic term for a bundle of sticks++