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Draco Malfoy has been Harry's enemy since the day they bumped into each other on the Hogwarts Express in Book 1. Draco tried to win Harry's friendship by encouraging him to stick to "the right sort" – i.e., rich purebloods like Draco. Harry's not a bigot or a snob, so he refused to shake hands with Draco, and the rest is history.
Draco is one of the leaders of Slytherin House; he is also (starting with Book 5) a prefect. Draco is in love with power and enjoys lording it over the Gryffindors as much as he can. Since he's a bully and a sneak, he looooves Professor Umbridge, who appoints him to the Inquisitorial Squad and gives him the right to take house points from his fellow students. Draco, of course, happily abuses this privilege:
Anyway, members of the Inquisitorial Squad do have the power to dock points ... so, Granger, I'll have fiver from you for being rude about our new Headmistress. Macmillan, five for contradicting me. Five because I don't like you, Potter. Weasley, your shirt's untucked, so I'll have another five for that. Oh yeah, I forgot, you're a Mudblood, Granger, so ten off for that. (28.10)
In other words, Draco doesn't even try to justify his punishments as anything other than outright dislike. And he calls Hermione a Mudblood – an awful name for a magical person with Muggle parents. So his prejudices continue to shine through.
Draco is a piece of work, but like Kreacher with the Black family, Draco has been made into the jerk he is by his family. His father, Lucius Malfoy, is one of the leaders of the Death Eaters under Voldemort. Draco boasts often of his father's connections: Lucius has ties to the Minister for Magic, and he is also supposedly close to Griselda Marchbanks, the head of the department that oversees OWL and NEWT exams for Hogwarts students. Draco claims, "Of course, it's not what you know [...] it's who you know" (31.43) with self-satisfaction.
Yet, for all of the bragging Draco does about his family and about his own personal power, he is also burdened by his family history. When Harry's interview appears in The Quibbler, Draco is furious that Harry has named Lucius as a Death Eater. And after Lucius goes to Azkaban for his ambush on the Department of Mysteries, Draco hisses, "I'm going to make you pay for what you've done to my father [...] You think you're such a big man, Potter. [...] You wait. I'll have you" (38.52). Draco carries on his father's politics and resentments without making his own choices about what he really thinks. He is a slave to his own background; perhaps if Draco had been raised by better (or at least less haughty and self-satisfied) people, he wouldn't be growing into the bully that he is in Book 5.