In JK Rowling's Harry Potter book series, the protagonists follow a format that is typical of the Joseph Campbell monomyth - a lowly boy finds out that he is 'special,' picks up friends along his quests, is taken to a magical new world where he has a greater deal of importance than his previous circumstances, and follows his destiny by defeating monsters and villains. However, Rowling's take on 'monsters' applies to Harry himself - those who are different are always persecuted against, even by those the reader may consider the heroes of the piece. Being different is what allows those with unique abilities to excel, and despite the opposing forces they often find themselves in, have a lot more in common than they realize.
First, Harry Potter is, in and of himself, a monster. He is often called a freak, and is placed in a separate class of individual from his peers - the other students of Gryffindor, and the Dursleys, as well as others. The beginning of the first book sees Harry being the outcast, the freak, persecuted because he is not wanted (he is the son of Vernon Dursley's more favored brother, making him ripe for punishment). Soon, even after he is introduced to the weird and wonderful world of the Potterverse, a place to which he is born as a celebrity. This comes as both a blessing and a curse to the varying denizens of the magical world he enters - the Weasleys, Dumbledore, all of his allies see him as "The Boy Who Lived," a child with great potential due to his unique nature. However, the Malfoys, the Ministry of Magic, and the forces of Voldemort see him as a threat - either to the status quo or of Voldemort's plan to take over the world of magic. Draco, in particular (and Ron Weasley at times) are envious of his celebrity and his special nature, hating the monster for stealing their thunder, more or less. The wizarding world looks at him with a mixture of awe and jealousy, and that is what makes him a monster, according to Rowling.
Some may define monsters as those who do not look like humans, especially in the Potterverse - there are many strange creatures that Harry and company have to contend with. However, despite their monstrous appearance, many of these are friends of Harry's, and their Otherness does not preclude them from being ethical or moral beings. Aragog and the other spiders are shown to be allies of Harry's; Remus Lupin, despite being a werewolf, fights by his side and is a member of the Order of the Phoenix. Many elves, like Dobby, are even shown to be peaceful and docile. Simply because a creature does not look human does not make it a monster, according to Rowling - this monstrous nature comes through in its actions.
Despite this ethical definition of monsters, it is very telling that many of the evil forces in the book do take the shape of monstrous creatures. The Basilisk in Chamber of Secrets is a deadly dragon-like creature; the Dementors are warped ghosts. Voldemort's own appearance is human enough to make him articulate and give him personality as a villain, but he looks very alien, with no nose, crooked teeth, no hair and sharp, pointy fingernails. Even the blue hue of his skin denotes that he is a monster. To that end, it is clear from Rowling's perspective that, even though not all beasts are monsters, the ugliest-looking ones are definitely evil. The most grotesque creatures are reserved for Voldemort's forces; most of Harry's magical beasts are owls, giants, hawks like Buckbeak, It can be concluded, then, that Rowling wishes to show that some creatures are good, but good guys typically follow the attractive-or-cuddly formula, unless otherwise presented for juxtaposition (as is the case with Fluffy or Aragog - or even Dobby).
The politics of difference between the different types of humans further demonstrates the fact that a 'monster' in Rowling's world merely constitutes the Other. As previously mentioned, the Dursleys and other Muggles consider Harry and other magical humans monsters, due to their eccentricity and, ostensibly, their power over them. Harry's repeated threats to Dudley and others to perform magical spells on them may seem like payback or defense to Harry, but in reality the Dursleys are scared stiff of him because of what they can do to him. Harry's abilities set him apart from the status quo - in fact, it turns him into a ticking time bomb of sorts.
The differences in wizard and human cultures are representative of race/gender/class distinctions that exist in the real world. House elves, Dobby for instance, are second-class citizens, working as indentured servants without recognition or agency, much like impoverished minorities and slaves. There are also questions of superiority regarding magic users and Mudbloods (those who are half-Muggle, half-magic user); the word "mudblood" itself is used almost as a racial slur to divide and segregate people like Hermione from the rest of the magic-using population. Draco, in his prejudice against mudbloods, is a party to a kind of institutional racism that has seemingly existed in his family for generations. The Weasleys seem to be lower-middle class citizens, the working class who have respectable jobs but are far from rich; the Ministry of Magic is painted as the elite that is often ignorant of the real goings-on outside their own perspective. In many instances, Harry has to lead a rebellion against the authorities of Hogwarts and the Ministry of Magic in order to achieve his own ends or save the day, indicative of many protest or underground movements. The differences found in these institutions are those of ideology and ethics; Harry feels as though he has to go outside the bounds of the law to get things done, when the law itself is not enough.
In the end, however, difference is viewed as a good thing in the Potterverse, as it allows enough diversity and uniqueness to defeat the forces of uniformity and conformity. Harry and co. are all unique in their own way - Harry is The Boy Who Lived, Hermione is a Mudblood, etc. - and their uniqueness helps to bring out the extraordinary bravery in the more mundane magic users (Ron, Neville). In essence, Voldemort wants to unite everyone under his own rule, while Harry's forces wish to celebrate the diversity of both the wizard and human worlds. In this way, the monsters are those who get in the way of that - regardless of the way they look.