If I have a chip on my shoulder, I am a little mad about something or someone. I may be quiet and I may not show it, but later, I will get mad more easily, and especially toward the person or thing that has upset me.

If you 'have a chip on your shoulder' about something, it means that you have a complex about it. For example, if a person is shorter, or balder, or from a lower social class than their friends and colleagues, they may feel inferior because of this.