To Bore: "The teacher began to bore me in History class".
To annoy: "My little sister annoys me". *Note that the spelling is "annoy", not "ennoy".
To Bother (This can have a couple of meanings): "The man living next door bothers me with his loud music". (He annoys me)
"I did not bother to do my homework last night/ I wasn't bothered about doing my homework last night." (I didn't care enough to do my homework last night).
"The girl and her friends got into bother with their teacher". (They got into trouble with their teacher).