Your bete noire is your enemy or nemesis. Batman's bete noire is the Joker. Your bete noire may be your nosy neighbor or your rival on the cheerleading squad.
Is there something or someone you hate more than anything else in the world? Then that's your bete noire. Superman's bete noire is Lex Luthor. In most action movies or TV shows, the hero has a bete noire. A bete noire can be a thing as well as a person: for a civil rights hero like Martin Luther King, racism was a bete noire. This is one of many terms borrowed from French.