Adapted from Structure of the Magazine, published by Nikola on Magazine Designing
ZineWiki defines a zine as "an independently- or self-published booklet, often created by a single person. Zines are customarily created by physically cutting and gluing text and images together onto a master flat for photocopying, but it is also common to produce the master by typing and formatting pages on a computer. The end product is usually folded and stapled. Zines can be printed and bound in any manner. Offset printing is a relatively common alternative to photocopying, though there is some controversy among zine writers as to whether professionally printed products may be defined as zines."
Zines can include all of the structural characteristics and identifiers of mainstream magazines, but the closer a zine comes to mainstream publication (such as gaining an ISSN and inter/national distribution) the more it becomes less of an independently or self-published zine. Zines can be mainstream production level quality, but far more often they are imperfect. It is possible for zines to be all of these things and/or none of these things. The very first zine is claimed to have been created in 1930, "The Comet" was a science fiction zine published independently. If you are interested in a timeline of zine history the Bingham Center Zine Collections at Duke University has a comprehensive outline.
Bitch (magazine) started as an independently published zine and now enjoys both international distributorship and a substantial online presence. Zines have a deep connection to the American hardcore/punk scene/movement going back to the 1980s and zines were used to share information about favorite lesser-known bands not being covered in mainstream popular culture. The Do-It-Yourself (DIY) movement is also closely connected to zines, as this format puts publication control in the hands of the individual or collective, rather than a formal publishing body.
A few characteristics that help in identifying zines: