There are many words to describe a person who thinks they are superior to others, each carrying slightly different connotations:
Arrogant
An arrogant person is characterized by an exaggerated sense of their own importance and a condescending attitude towards others. They often boast about their achievements and look down on those they perceive as inferior.
Conceited
Similar to arrogance, conceit refers to an inflated sense of self-worth. However, conceit often focuses on physical appearance, possessions, or social status rather than intellectual or moral superiority.
Egotistical
An egotistical person is excessively preoccupied with themselves and their own needs and desires. They crave attention and admiration and often lack empathy for others.
Haughty
Haughty individuals display an attitude of superiority and disdain for others. They often come across as aloof and dismissive, believing themselves to be above the common crowd.
Pompous
A pompous person is characterized by excessive self-importance and a tendency to show off their knowledge or status. They often use elaborate language and grand gestures to impress others.
Snobbish
Snobbishness refers to a disdain for those perceived as lower in social status or lacking in cultural sophistication. Snobs often look down upon others who do not share their tastes or interests.
Supercilious
Supercilious individuals are characterized by an arrogant and condescending attitude. They often express their superiority through patronizing behavior and dismissive remarks.
The specific term you choose to use will depend on the specific nuances of the person's behavior and the context in which they are described.