The label narcissist is used in reference to a person displaying narcissistic traits. The medical diagnosis is actually Narcissistic Personality Disorder (NPD). To meet this diagnosis it is necessary that a person meet at least 5 of 9 criteria.
According to the American Psychiatric Association's Diagnostic and Statistical Manual of Mental Disorder, Fifth Addition (DSM 5) NPD is defined as a Cluster B personality disorder comprising of a on going pattern of grandiosity (in fantasy or behavior), a constant need for admiration, and a lack of empathy, beginning by early adulthood and present in a variety of contexts, as indicated by the presence of at least 5 of the following 9 criteria:
A grandiose sense of self-importance
A preoccupation with fantasies of unlimited success, power, brilliance, beauty, or ideal love
A belief that he or she is special and unique and can only be understood by, or should associate with, other special or high-status people or institutions