A mixture of the words Faggot (as a general insult, not homophobic) and critic. This term can be applied to anyone who likes a certain genre of something and will mercilessly tear a new asshole into anyone who likes other wise.
More common in music and film tastes, books have been known to be subject to a 'Faggic's' hatred too.
Person 1: Dude, you have to hear band name's new song! It's the bomb!
Person 2: I fucking hate band name! They're so emo/chav/pop/punk, all of which are terrible. I'm incredibly singleminded and will ONLY listen to the one band. Everything else is shit.