It can also mean that the person has a grievance about something. Something bad has happened to him, and he might believe it was someone else’s fault. This kind of person is constantly angry about it that it affects his behaviour. In the person’s mind, he is showing how tough he is.
You can also use have got. You can also add adjectives to describe the size of the chip.
I can’t stand working with Bill; he’s got such a massive chip on his shoulder. He’s so unpleasant.
…Keira Knightley has admitted that forsaking university in favour of a film career has left her with a “chip on her shoulder” over her lack of education. . . . She now battles to prove to herself – and those around her – that she has brains.
Excerpt from The Telegraph - U.K.
Success never made that chip on his shoulder go away.
Excerpt about Roger Ailes, Fox News Chief, from The New York Times
James “Sawyer” Ford - a character on the TV show LOST- started as the classic rogue with a shady past who parades around with a constant chip on his shoulder.
Excerpt from IGN.com
2. "The only way to satisfy an entire Harem in under 5 minutes is to employ a chip."