"Having a chip on one's shoulder" is a pretty weird way of saying that someone is spoiling for a fight or has, as we say these days, an attitude problem. I guess it's a tribute to the capacity of human beings to accept bizarre figures of speech in everyday conversation (Is there really "more than one way to skin a cat"?) that some of these phrases have lasted as long as they have.
And "having a chip on one's shoulder" has actually been around for quite a while.
The earliest printed instance of the phrase listed in the Oxford English Dictionary comes from the Long Island Telegraph newspaper in May, 1830. This citation also provides what is probably a good explanation of the origin of the phrase: "When two churlish boys were determined to fight, a chip would be placed on the shoulder of one, and the other demanded to knock it off at his peril." (The "chip" was, in that age of wood stoves, most likely a chip of wood.) Evidently this belligerent ritual of childhood was sufficiently widespread at the time to become a grownup metaphor for combativeness, as it has been ever since.
