"What is a zine?"
In a nutshell, a zine is a booklet, usually made by hand.
A zine (pronounced ZEEN, not ZIGHN!) is a "little magazine." Unlike magazines, zines are made by one or a handful of individuals, not a large corporation, and they are generally made by hand, either through analog or digital means.
A very popular style of zine is the 8-page zine, which is folded from one sheet of paper. But there are many styles of zines!