Admitting I read dōjinshi (Japanese fan made and published comics akin to fan fiction), much less commenting on them, is not something I do very often. While there are some very good non-hentai dōjinshi, they are not anywhere as easy to find (the series in which I have seen the most non-hentai and good dōjinshi is Slayers). The reason for this is because most dōjinshi are of the Hentai variety, at least most dōjinshi that get posted on the internet are, and have either no plots or very weak plots. There are very good quality dojinshi, even of the hentai variety, quite a few mangaka started out as dōjinshi writers or write dōjinshi on the side. Ken Akamatsu (Love Hina), Rikdo Koshi (Excel Saga), Kiyohiko Azuma (Azumanga Daioh) and Masaki Kajishima (Tenchi Muyo) all wrote dōjinshi prior to becoming mangaka. Some still do like Masaki Kajishima, he writes mostly Tenchi Muyo dōjinshi that is supplementary to the series.
Writing fan made comics is not unique to the Japanese. The biggest difference is that they are more illegal in some places, like the USA, because of copyright laws. The Japanese copyright laws are not as stringent and the industry allows the publishing of dōjinshi (they even have large conventions for dōjinshi).
Some particularly popular anime/manga spawn webcomics (which are for all practical purposes the same as dōjinshi, just not produced in Japan). Two examples are: Sailor Ranko (a fukufic i.e. Ranma/Sailormoon crossover - an alright but not great fan comic) and Slayers Hamlet (Shakespeare's Hamlet with the Slayers cast - a most excellent but sadly not updated in several years webcomic).
A few good dōjinshi are:
-Evangelion Re-Take, which has both a hentai version and a non-hentai version. This has a very good plot.
-Blackdogs Sailormoon dōjinshi, which are hentai and have very little plot but are drawn very well. They can be found at Sailormoon.biz along with a few hundred other sailormoon dōjinshi.
-There are quite a few Slayers dōjinshi that are not hentai, mostly WAFF (warm and fuzzy feeling). I 'd link to them but most of the links to them that I have are non functional and I know not where to find them anymore. The only links in my bookmarks that still works are this Fortuncity site Isabel's fanworks corner, but it registers as a dangerous site according to firefox, QP House which has a few dōjinshi as well as summaries of the Slayers Novels, and Slayers Doujin fan share which has quite a few. I also know of this site which has Hentai and Yaoi pairings but the hentai in general isn't that good and Yaoi disgusts me.
- The Original Dirty Pair / Lovely Angels Shrine has quite a few Dirty Pair things like an entire novel and several dōjinshi the best though is listed under fanmade comics and is a Ranma/Dirty Pair crossover.
-There are a few good Ranma dōjinshi but I don't have a list of links at this time, the sites that had them generally do not have them anymore.
I request of any reader of this to suggest some good dōjinshi in the comments (Hentai variety is acceptable, but please mention that it is) and links to it would be appreciated.
Turtle Paradise Scanlations, that has several doujinshi including a couple Slayers doujinshi (as well as several Slayers manga scanlations). I've yet to read any of those manga or doujinshi so I can't say how good they are. Though I have read and own several other Slayers manga and enjoyed them so hopefully I'll enjoy these as well. I'm somewhat surprised I hadn't found them earlier but seeing as to how I haven't been reading or looking for Slayers stuff recently I'm not that surprised.
The website Realm of Sailor Energy has some amusing comics on it.
The Fancomic Central livejournal has links to numerous fan comics.