Whether found as a child or discovered later in life, all Harry Potter fans are connected by their love of the magical world and it's inhabitants. While the mythical creatures, enchanted land's, and witchcraft/wizardry are each solid points of facination; it is the characters to whom we come to know the best.
Much like any other literary or film success, the genius behind the Harry Potter franchise is the connection which readers and viewers come to share between one another, and to the characters themselves. By creating interesting, relatable, and at times flawed characters; fans are able to fully imagine the people and circumstances of that world. They feel the character's triumphs, failures, sorrows, and joys. They find the intricate aspects of each character's design, and relate them to their own. So that at times, they experience the suspense of decision making, the fear of danger, and the marvel of this alternate world- as if they were truly there themselves.
With such an array of characters, it can be difficult to not become invested (positively or adversely) in their lives. With each read or viewing, more possibilities are revealed as to their possible motivations and true feelings. "Good" characters (like Dumbledoor) can be shown to have more flaws, while "bad" characters (like Draco Malfoy) can be revealed to have more of a depth of uncertainty and dichotomy of reasoning. Yet regardless of whether the above characters are to be commended for their positively intented actions or damned for their negative ones, I think we can all agree that of any of the antagonists; Umbridge is one of the most evil. As her motivations are based solely in the (racist) beliefs of blood purity; targetting muggle-borns, werewolves, centaurs, and hybrids. Not to mention her actions of child abuse with the blood quills, and how she was seemingly unaffected by the Slytherin locket Horcrux (or at least in a negative way).
Speaking of child abuse, Dumbledoor and to a lesser extent much of the Order of Phoenix are to blame for much of Harry's mental health issues. While the books and movies are never terribly detailed in the exact extent of his mistreatment at the Dursley's; it is important to remember that for much of his life, he lived in a small cupboard under the stairs (which was included in the address of his Hogwarts letter). He was made to do laborous chores from a young age, punished for magic, bullied by Dudley and his gang at school, and had basic needs (food and clothing) largely ignored. If Dumbledoor knew of the extent of these conditions and ignored them for the sake of the "greater good", then when compared to Tom Riddle's upbringing and future as Voldemort- it is a wonder that Harry's life did not have a similar outcome; as any early form of affection he'd been exposed to was limited to Miss Fig with her cats and old cake, or to random strangers on the streets thanking him. Surely this is not what James and Lily would have wanted? Yes, through the family bond of blood and Lily's love he was protected from Voldemort, but at what cost? Oppositely, what would have happened if Voldemort had been adopted, not made to go back to the orphanage in the summer, or not concieved under the influence of a love potion? If Peter had never become frinds with Remus, Sirius, and James? If Sirius had remained secret keeper, or had taken Harry from Hagrid instead of confronting Peter? If Snape had never become a death eater, or overheard the prophecy?
It is these deeper connotations of actions and motivations; of questions without answers, that fuel the displayed personality of the characters; as well as prompting further contemplation by die hard fans and critics alike. Because even though the books and films are based largely in fantasy, the more basic forms of character and emotion are very realistic to everyday life. It is because of this that their words can hold so much meaning, to so many people; that their world is so alluring in its familiar, yet so different way to our own. It is each of these qualities and more, which bring the world of Harry Potter to life in the minds of readers and viewers alike, becoming a part of them even more.