“Harry Potter? Who's that? All I saw was the noble scion of the Malfoy family and some half-blood.”

Dedicated blood-purists like Blaise Zabini don't always hate half-bloods, but they particularly hate Harry because Harry is the offspring of James Potter (the last descendant of a highly respected pure-blood family) and Lily Evans (whose parents weren't even magical). So, to them, Harry represents the destruction of an entire lineage.

(Most people in this society don't have such extreme attitudes, but they still look down on Harry because ze was raised by muggles.)