In Pygmalion, middle-class morality is the expectation of proper behavior for those belonging to the middle class. This term is developed into opposition toward the looser moral standards of the lower classes.
Eliza's father, Alfred Doolittle, goes into the subject when trying to squeeze money out of Higgins when he learns Eliza is living in his house. Because Alfred is poor, no one expects proper behavior from him. He lives with a woman without the benefit of marriage. He drinks excessively. He does not work. He is perfectly comfortable conning people out of money or even "selling" Eliza to Higgins so he can have more drinking money. For Alfred, his lower-class standing equates to a freedom that the upper-class Higgins or even just middle-class society cannot have because to do so would compromise their privileges.
By the end of the play, Alfred is forced to adopt this middle-class mode of morality because of his newly procured money and position. Now he has to marry his longtime girlfriend just to keep up with appearances. While he enjoys the comforts wealth brings him, Alfred mourns his lost freedoms. He even goes as far as to call living out of wedlock with a woman as "the natural way" as opposed to "the middle-class way" of officiating a relationship with marriage.