This is quite loaded question, and I'm sure I could go on for days!
On a personal and somewhat petty level, I get annoyed by certain sounds: nails clipping, silverware clanking on plates, gum chewing/popping, TV when I'm trying to sleep. I've been told I have acute hearing, and that's why these things bother me so much.
As a teacher, there are plenty of things that annoy me. Students grooming (doing nails, brushing hair, etc.) in class drives me absolutely insane. I also can't stand it when students use phones when I have asked them to put them away.
More importantly, I am very much annoyed when students have an "I don't care" attitude toward their education. We have come to a time when people take their education for granted and don't realize how lucky they are to have a free quality public education- it wasn't always that way for everyone in this country, and it's still not that way for some in other countries. But that appreciation has been lost.
I also get annoyed when people try to guilt-trip me. That never makes me want to do something.