When one twin is clearly more attractive, cool, or likable than the other twin. After suffering through a lifetime of people asking where the "hotter" or "funnier" twin is, the less desirable twin inevitably forms a "chip" on his or her shoulder, thus making the chip twin even less enjoyable to be around.
1.
Party host: "Where is Andy? I love that guy. He's hilarious."
Chip twin: "Why is everyone always asking me where Andy is? How should I know? I don't know where he is all the time! I'm just as funny as he is!"
Party host: "Listen chip twin, you're not invited unless you are with Andy. Robin didn't go around trying to fight crime on his own."
2.
Guy at bar: "So how do you two know each other?"
Chip twin: "We're twins."
Guy: "No, seriously. How do you know each other?"
Chip twin: "We are twins."
Guy: "Come on. Seriously, how do you know each other?"
Chip twin: "You're an asshole."
*Twins walk away*
Guy: "What did I say?"
*Guy 2 walks up*
Guy 2: "Fucking chip twins, man."