Not much! Ideas are probably the most important part, but once you've got that down paper, scissors, glue, writing instruments, photos or other collage materials, and a copy machine for quick duplication are probably the most important. You can make a zine as elaborate or as simple as you'd like.
The following useful materials can be checked out from the circulation desk:
Probably the easiest way to get started is with a one-page mini zine. If you use 11 x 17 paper, the result is a postcard-sized zine that can be easily distributed through the mail. Here's a couple tutorials:
If you'd like a few more pages to work with, this tutorial results in a 16-page zine that has some interesting twists and turns:
There are also zines that are more book like, made with 8 1/2 x 11" pages folded in half. There are templates for helping you with page numbering, such as these from Anatomic Air Press.
If you're interested in that style, the following video is a really useful guide: