Something that's annoying is upsetting or irritating or otherwise troublesome. When your mom asks you ten times in a row when you're going to clean your room, she's being annoying. But your messy room might also be annoying!
Annoying really is a tiresome word, as its roots imply: it comes from the old French word anuier, meaning "to weary or vex," and from the Latin inodiare, meaning — even worse — "to make loathsome." Think about that next time you do something annoying. And don't forget, it's easy to be annoying without even knowing it, as Mark Twain understood when he said, “There is nothing so annoying as to have two people talking when you're busy interrupting.”