Which theory of personality is most concerned with understanding people’s unconscious motivations?

The answer is: Psychoanalytic

According to psychoanalytical theory of personality, the minds of every human being is divided into conscious and unconscious thought. This unconscious thought often represent our true nature and can influence our behavior without we even realizing it.

So, in order to analyze the patients, psychoanalytic therapists often use various forms abstract tests (such as the inkblots) in order to find out unconscious motivations that might exist within the patients.