A "pet peeve" is a particular annoyance or irritation that bothers a person, often to an exaggerated degree. It's usually something that may seem minor or trivial to others, but for the individual, it can provoke a strong emotional response. Pet peeves can vary widely from person to person; what annoys one individual might not bother someone else at all.
The term "pet peeve" combines "pet," meaning something that one has a special fondness for, and "peeve," which is derived from the verb "peevish," meaning to complain or whine. The phrase became popular in American English in the 20th century.