A quick taxonomy of fanfic (from someone who spent way too much time in HP fandom in college):

fill-in-the-blanks: This is a story that tells parts of the original work's backstory. The custom is to try scrupulously to maintain canon compatibility -- ie ensure that your work, and the original works, could be presented as a unified whole without contradiction. Fernwithy (http://www.sugarquill.net/index.php?action=profile&id=50...) writes beautifully in this mode, is very careful about maintaining consistency, and has only started writing stories set in the future of the series since the 7th book closed the canon up.

The future fic:
This is the sort of thing authors do because they're impatient for the next installment of canon. They try to write the next installment themselves. Again, consistency is considered important. Usually there's some wish-fulfillment, not so much in terms of self-insertion, but in terms of the author taking the story where they wished it would go, especially by pairing two characters the author likes together.

After the End(http://www.sugarquill.net/read.php?chapno=1&storyid=619) and The Letters of Summer(http://www.phoenixsong.net/fanfiction/story/53/) are both good examples.

Once the next installment of canon comes out, the future fic is usually retroactively labelled AU, or alternate universe.

The single-point-of-departure AU:
"For want of a nail, the horse was lost..." -- the author picks a very specific time and place in canon in which something goes differently, and then tells the story from there. Even here, consistency is prized, unless you can justify how the change arose from the point of departure. I haven't read much in this mode, so I don't have any immediate suggestions.

A strangely specific variant of the single-point-of-departure is the Peggy Sue, in which a main character travels back in time, inhabits the body of his/her young self, and redoes everything using their knowledge of future events. Harry Potter and the Nightmares of Futures Past (http://www.fanfiction.net/s/2636963/1/Harry_Potter_and_the_N...) and Harry Potter and the Wastelands of Time (http://www.fanfiction.net/s/4068153/1/Harry_Potter_and_the_W...) are both examples. Peggy Susie (http://www.fanfiction.net/s/5731071/1/Peggy_Susie) is a one-shot parody of this genre by MoR's author.

The crack-fic:
The author takes the story miles into left field, usually played for laughs. How Hogwarts Became a Nudist Colony (http://www.fanfiction.net/s/2179136/1/How_Hogwarts_Became_a_...) or Potter Puppet Pals (http://www.potterpuppetpals.com/newppp/channels/TroubleAtHog...) are prime examples. No rules here but the rule of Funny.

MoR doesn't fit neatly into any of these categories, but starts out sounding a lot like it's going to be a single-point-of-departure AU (the notes at the top of the first chapter have now been edited to specifically disclaim this). MoR is an example of, well, taking another author's universe, using the bits you like, and changing whatever you need to change to tell the story you want to tell, in the manner you want to tell it. It shares this "genre" with most comic reboots, Homer, Milton, the Bible, the brothers Grimm, Disney -- basically a huge chunk of western culture. But it still doesn't quite fit into the set of genres commonly seen in HP fanfic.