To be thin-skinned is to be sensitive, especially to criticism and slights. Thin-skinned people are quick to take offense.
If your skin is literally thin, soft, and tender — like a baby's — than you're going to be hurt more easily if you get scraped or cut. Similarly, a thin-skinned person gets emotionally hurt more often than most people. If you're thin-skinned, you take criticism, rejection, disappointment, and failure very hard. Being left out of anything could be perceived as a major insult. If you forget a thin-skinned person's birthday, they’ll probably hold a grudge.