If you have an exaggerated opinion of yourself, believing you're smarter, more attractive, or just generally better than everyone else, you're stuck-up. If you talk about nothing but the awards you've won, and your friends might think you're stuck-up.
You can also describe a stuck-up person as arrogant, snobby, or conceited. The adjective stuck-up is informal, but it's a great way to talk about someone who brags about himself and looks down on just about everyone else. Some experts guess that stuck-up comes from the idea of "having one's nose up in the air." We do know that it first appeared in print around 1830.