It means someone is easily angered and always ready to “fight” or “argue” about something. The saying was started in the 19th century in the US. People who wanted to get into a physical fight would carry a chip of wood on their shoulder and dare others to knock it off. When someone knocked it off, it was the start of a fight.

If a person feels inferior or is very defensive and hostile about some issue and ready to argue or even fignt at the least cause, we say that that person has a chip on his shoulder. It comes from a practice of one man putting a chip of wood on his shoulder and daring another man to knock it off. If the second man pushed or tapped the wood, it would fall and that would mean the second man accepted the challenge and wanted to fight also. As you can imagine, it would not take much to knock a piece of wood off your shoulder. The slightest tap would do it. So if a person "has a big chip on his shoulder", it would not take much to make him angry.

There's a bit of history behind this phrase, but basically it is a metaphorical phrase that describes someone who is nursing a grudge that readily provokes anger; someone angry over something that happened in the past.

A person has a bad attitude