Chip, in this common English idiom, actually refers to “a small piece of wood.”
The phrase, to carry a chip on your shoulder, means “to carry a grudge”—to feel angry or desire to fight because of past unfair treatment. For example, people often carry a chip on their shoulder for a past lover or parent who mistreated them:
Don’t carry a chip on your shoulder; instead, “forgive and forget.”
Kevin carries a chip on his shoulder for the kids who made fun of him in school.
The phrase originated in a 19th century American tradition, in which one boy might challenge another to fight by placing a wood-chip (a small piece of wood) on his shoulder—in this way, daring another boy to knock it off.
Therefore, the phrase implies that a person wants a fight, but will not start it; instead they passively dare other people to start a fight. If someone is always rude to you for no reason, you may think that they have a chip on their shoulder, based on some past experience with you, or people associated with you. In other words, to carry a chip on your shoulder is considered highly neurotic and is often used in a pejorative sense. Although, the phrase is sometimes used to refer to someone in a more positive sense, where the person who carries a chip on their shoulder is being praised as someone who is tough and doesn’t back down from a challenge—not afraid to take on anything that comes their way:
Gabe plays football like he’s carrying a chip on his shoulder; he treats every play as an opportunity to prove his doubters wrong.