Originally used in opera companies, "prima donna" is Italian for "first lady." The term was used to designate the leading female singer in the opera company, the person to whom the prime roles would be given.
Legendarily, these "prima donnas" were often egotistical, unreasonable and irritable, with a rather high opinion of themselves not shared by others. Thus the modern use of the term to mean a vain and temperamental person who, although irritating, cannot be done without.
See also