to have a grievance or to hold anger about something.
Examples:
- Danny seems to have a chip on his shoulder. I wonder what happened.
- Yes, I do have a chip on my shoulder. I am tired of you coming late to the office every day.
Origin: In the old days, when a person was angry with someone, they would put a wood chip on their shoulder. If the other person knocked it off, it meant that they were ready to fight.