In short a zine is a self published pamphlet, often reproduced on a photocopier, for small scale circulation. They are noncommercial and nonprofessional having been produced, published and distributed by their creators.
You have most likely read a zine before. They come in numerous forms: Football fanzines, political zines, feminist zines, sci-fi zines, music zine, art zines, comic zines, religious zines and photo zines to name just a few. Basically a zine can be about what every you want.
Zines are symbol of independence, a staple of DIY culture and a voice of the counter culture.
With all that in mind it is your definition of a zine that is this most important. Not mine!