Harry is normally portrayed in fanfiction as 'Light', 'Dark' or 'Grey' in reference to what type of magic he uses, with the additional note of 'good', 'evil', 'but not evil' or, rarely, 'neutral' to indicate his alignment in the war. The 'Dark!Evil!Harry' type normally has him either joining with Lord Voldemort or overthrowing him. 'Light!Good!Harry' is almost always on the side of Dumbledore and the Ministry. 'Dark!but not evil Harry' tends to be shunned by the 'Light', either the Ministry and/or the Order, because of his use of 'Dark' magic and/or possession of 'Dark' creature genes, but he still helps them against Voldemort. 'Dark!neutral' or 'Light!neutral' Harry often creates his own faction in the war and overthrows all of Magical Britain to create a utopia for all magical beings therein.
Harry is one of the most popular people to give a magical creature heritage to. This includes, but is not limited to, Veela, dark Veela, Elves/High Elves, werewolves/other were-creatures, vampire, dragon and fae. If it is a 'Dark' creature then it usually results in him being shunned by others, unless he manages to use his greatly improved physique and newfound magical reserves (which he almost always gains no matter what the inheritance is) for the sake of good.
Whatever type of fic it is, there is almost always a Prophecy to be considered. Despite the fact that there is a canon Prophecy about Harry and Voldemort, many Suethors feel the need to make up another, usually poorly rhymed, Prophecy that states that Harry and the Sue will be the ones to defeat Voldemort. Sometimes the Sue will be substituted for a canon character - usually Harry's love interest - but the same rule still applies.
If the fic is set in Hogwarts, there will always be problems with Malfoy, snarkiness from Snape, a mysterious (and doomed) DADA professor and a showdown with Voldemort at the end of the year. Ron, Hermione and Ginny may or may not stand by him, depending on what choices he makes about his life.
If the fic is set after Hogwarts, then Harry is normally an Auror, a professional Quidditch player, or a Healer, who is called on to stop the next great evil, usually Malfoy or Voldemort's son/daughter. If the case is the latter, then she is normally a Sue and Harry will fall in love with her and she with him, following which either she will be redeemed or he will join the dark.
Another common story line is the raped/abused Harry who has to be taken care of and put back together by a caring person, be it a Sue or a canon character. This normally ties in with one or more of the above plot types, and quite often features Mentor! or Fatherly!Snape. The Dursleys are the typical source of the abuse, but on occasion, it can be as a result of capture by Death Eaters.
Harry doesn't have any real familial relationships as his parents died at Voldemort's hand when he was just one year old and the Dursleys didn't care very much for him. He has a pseudo-family in the Weasley clan, and Sirius Black treats him as if he is James Potter for the short time that they know each other.
Canonically, Harry is romantically linked to Ginny Weasley and, briefly, Cho Chang. A surprising number of fanfiction authors respect this and pair Harry up with one of those two. But Harry is also frequently paired with Hermione Granger, Luna Lovegood, Nymphadora Tonks, Narcissa Malfoy, Bellatrix Black or a Mary Sue. In slash fics Harry's most common partners are Draco Malfoy, Severus Snape, Voldemort/Tom Riddle and Blaise Zabini.
If the fic is a creature!Harry type then the pairing is usually born of a need to find his destined mate within a certain amount of time or he will die a horrible and painful death. The normal recipient of this is either a Sue, Malfoy or Snape.
Harry suffers from 'Main Character Attractiveness Disorder'.