The first major complication in Harry Potter and the Half-Blood Prince occurs when the reader observes Snape secretly allying himself with the Malfoy family. This creates tension because the reader is used to seeing Snape as an anti-social but generally good character, and pays off at the end when Snape kills Dumbledore.
The second major complication also involved Snape; he is finally allowed to teach Defense Against the Dark Arts, which makes class harder for the students. Harry in particular is discomforted because he and Snape never got along; he is helped by a mysterious book which turns out to be Snape's personal potions textbook.
The third major complication comes when Harry realizes that he has feelings for Ron's sister, Ginny. This conflicts with the trusting friendship he has with Ron, and also with his need to keep other people at a distance because of his dangerous life.