Hank "Rusty" Truman asked: In what ways does Azula annoy Zuko, being his little sister and all?
She just acts like the typical brat, following him around when he wants to be alone and ignoring him when he wants her company, mocking him to get on his nerves whenever she has the chance, teasing him at inappropriate times (be it regular sister teasing or going full "incestuous brat that loves seducing her brother in public" mode), getting him all worked up and needy and then not letting him touch her, being intentionally bitchy to his friends, making sarcastic comments at his expense, calling him by embarrassing nicknames in front of others (if you think he is mortified when she calls him "Zuzu" you should see his face when she calls him "love muffin")...
Azula is a professional in bratiness - good thing Zuko is the best brat-tamer.