It's a peeve (annoyance) that you enjoy having... like it was your pet.

something that really annoys you

Peeve = 'annoyance.' Pet (adj.) = 'Particularly cherished or indulged' So, a 'pet peeve' is an annoyance you particularly cherish. Like: "Editing your post after someone else replied is a one of my main pet peeves."

My pet peeve or pet hate is when car drivers cut you off or don't indicate.

The same question was appeared on the answer.com, it is possibly your favourite minor complaint. Answer: It's when someone, or somethings, actions really irritate you and you just can't stand it. For example, I hate it when a person goes to turn while driving, and they don't turn on their blinker.