If someone is supercilious, he or she is arrogant. You could use the term stuck up. A person who has an exaggerated self-opinion (someone who thinks he's a lot better than anyone else thinks he is) can also be described as supercilious.
What does the word supercilious mean?
This word is used frequently in literature.
From Oscar Wilde's The Picture of Dorian Gray:
"I am told, on excellent authority, that her father keeps an American dry-goods store," said Sir Thomas Burdon, looking supercilious.
From The Mayor of Casterbridge, by Thomas Hardy:
The hay-trusser, which he obviously was, nodded with some superciliousness. Looking towards the village, he continued, "There is something going on here, however, is there not?"
From Babbit, by Sinclair Lewis:
Now they were men and women of the world, very supercilious men and women; the boys condescended to Babbitt, they wore evening-clothes, and with hauteur they accepted cigarettes from silver cases.
And from Theodore Dreiser's Sister Carrie:
There was nothing bold in her manner. Life had not taught her domination — superciliousness of grace, which is the lordly power of some women. Her longing for consideration was not sufficiently powerful to move her to demand it. Even now she lacked self-assurance, but there was that in what she had already experienced which left her a little less than timid.