"Zine" is short for fanzine. For all intensive purposes, a zine is a cheaply-made, cheaply-priced publication, often in black and white, which is mass-produced via photocopier and bound with staples.
Most zines revolve around a music scene of some sort, but others are dedicated to artwork, poetry, cartoons, editorials and short stories. Because zines do not have any sort of corporate backing, they are very rugged, individualized, and much more charismatic than larger, more popular magazines whose content is often dictated by their advertisers.
Have you read the latest issue of my zine? I have an in-depth interview with the singer from Pernicious Crotch Fungus!