"WASP" is an acronym for white Anglo-Saxon Protestants. When someone calls someone a WASP today, they usually mean to imply that the person is uptight, reserved and/or elitist, regardless of their background. The stereotype of the uptight, upper class WASP is based on the assumption that white Anglo-Saxon Protestants are both genetically (white Anglo-Saxons) and religiously (Protestant) predisposed to acting like, well, snobs.
The term "WASP" was made popular by E. Digby Baltzell in his 1964 book, "The Protestant Establishment: Aristocracy & Caste in America." Baltzell used the term to refer to the American cultural, political and economic elite--preppy, Ivy League and country club types. These days, the label is used much more loosely to refer to people--almost always white, sometimes Protestant--who fit the behavioral stereotype of a WASP. The Anglo-Saxon part of the term--which refers to descendents of Germanic and Danish people who settled in the British Isles and are by definition white--has become irrelevant.