Zines
Fanzines are amateur made magazine-like books which contain fan fiction, fan art, poems, songs, reviews, and/or any other material developed by fans based on a fandom such as Harry Potter. Because editors put a lot of time (and money) into putting together a fanzine, they edit the stories they receive to ensure that spelling and grammar is correct, that there aren't any plot holes, and that stories are finished before they accept them. Although the quality of the material varies depending on how strict the editor is, fanzines tend to contain fic and art that is of better quality than most stuff found online.
Now, because it costs money to put together a zine and to print it, a fee is charged for the zine but only that which it cost the editor to create the zine in the first place (in fact the editors often lose money in the process) and to send it to the buyer. In order to ensure that people are willing to buy the zines, any stories or artwork that is accepted by an editor is generally not allowed to appear online for about a year (this time restriction can vary depending on the editor and whether or not the zine sells out, if it does, the fic and art can often be posted online sooner). Contributors normally get a free copy of the zine if their work is long enough (a story of over five pages and three pieces of art is the norm but again, it varies from editor to editor) or a reduced price if not.
As far as I know, there is currently only one zine that accepts Harry Potter material. And, unlike most fanzines, it is a multifandom zine (which means it accepts material from fandoms other than Harry Potter as well). If anyone is aware of another zine that accepts Harry Potter material, inform me (valeriev84@hotmail.com) so that I can add it to my list below.
Websites with information on zines or with zine material:
Millennium is the only fanzine, that I know of, which prints Harry Potter material. It is, however, a zine that started out as a Star Wars zine and now accepts Lord of the Rings, Pirates of the Caribbean, and Star Trek material as well as Harry Potter and Star Wars. It is a publication of Falcon Press.
The Fanzine Archives is an non-profit organization that seeks to ensure that fanzines continue to circulate among fans and are preserved. These archives are federally recognized and contain thousands of zines, several hundred of which can be rented.
The MediaWest*Con is a huge convention held every year on Memorial Day weekend in Lansing, Michigan, that is attended by people from all over the world. It's focus is on science fiction fanzines, but there is also an art auction, the Fan Quality Award, a music video room, a dealer's room, a dress-up contest, role playing... and much more.