Annoy and Irritate

Annoy

If someone annoys you, he is troublesome and causes you some slight bother. Annoyance is anything that cause you a slight problem, and it is described as annoying.
Ex: I know you've heard it a million times, but it's true that little brothers are the most annoying creatures on Earth. My little brother, for example, never stops talking.
Ex: My boss annoys me when he reminds me to clean the glass countertops every day. I've worked for him for six months, so I know the glass needs to be cleaned several times a day.
Get under one's skin is an expression that means someone has annoyed another person who cannot ignore it. If someone is annoying you, they are bugging you. After all, bugs are annoying.
Ex: Alec's clicking his pen constantly during the final exam really got under my skin. In fact, I was so upset that I got a bad mark.
Ex: People talking on their mobile phones in the supermarket checkout queue really bugs me. They make their private business public, and they hold up the queue.

Irritate

To irritate is to behave in a way that makes another person impatient or annoyed. What that other person feels is irritation.
Ex: I was irritated when the customer service representative took her afternoon break just as I got to the front of the queue because I had other errands to run before returning to work.
Ex: John's irritation increased when he could not get a human on the phone after he called his bank for information on his account.
If you irritate someone you might ruffle his feathers or get in his hair. An irritant is a substance that causes a physical reaction when skin comes into contact with it. If you annoy someone for a long time, he may become irritable.
Ex: Students who arrive late to Mr. Mortimer's class really ruffle his feathers causing him to pace around the room. It takes him a while to calm down.
Ex: Jenna has a sensitivity to some metals. Sterling silver jewellery, for example, acts as an irritant and causes a rash on her skin.