Definition
How to best express annoyance in German largely depends on whether it is done on purpose or not.
The words nerven and ärgern can both be used, and the reflexive sich/mich/dich/ihn etc. changes the meaning.
Examples
Er nervt mich! He’s annoying me! (not on purpose)
Die Musik nervt! The music is annoying!
Ich bin genervt! I am annoyed!
Ich ärgere ihn gerne! I like bothering him. (on purpose)
Die laute Musik ärgert mich! The loud music is bothering me.
Ich bin leicht verärgert/genervt. I am easily annoyed.
Lass dich nicht verärgern! Don’t let yourself be annoyed.
Good to Know
The word irritieren DOES NOT mean “to irritate”. In German, irritieren has the meaning of “to confuse” or “to distract” rather than “to annoy”.
