degree: a course of study at a college or university, or the qualification given to a student after he or she has completed his or her studies:
Examples:
She has a physics degree/a degree in physics from Edinburgh.
(mainly us) She has a bachelor's/master's degree in history from Yale.
(UK) "What degree did you do at York?" "Geography."
scholarship: an amount of money given by a school, college, university, or other organization to pay for the studies of a person with great ability but little money:
Examples:
He has dedicated his life to scientific research.