'Peevish' doesn't mean 'annoying'. It's a very old-fashioned word used to describe person who is easily irritated. You can forget about this word completely - you will probably never hear anyone use it and you will certainly never need to use it yourself.

A pet peeve is a minor annoyance. Often a pet peeve is something that you might find annoying but someone else may not care about. Something that is annoying is generally bigger.