Pygmalion explores class conflicts: conflicts between an ideology that believes class attributes are in-born (nature) versus Higgins's belief that he can teach class to Eliza (nurture) and pass her off as a lady. By creating a character in Eliza who quickly learns to speak and act like a higher class woman (though not without some comic blunders), Shaw punctures the idea that one class is naturally superior to another.
A second conflict occurs in class values: when Eliza's father comes into money, he finds the shift in expectations burdensome. He had adapted to a set of values in which he wasn't expected to marry, nor was he expected to take responsibility for other people. Once he has money, he complains of being expected to marry his partner and help out poorer relatives, as well as to look and behave respectably by middle class norms.
Finally, Eliza's self-supporting, if meager independence as a flower seller comes into conflict with middle class expectations that women be ornamental but largely useless. She attains the attributes of a middle class woman without the income, which means she will be destitute if she doesn't marry. Shaw shows that lower class people can rise up the class ladder but at the same time questions the advantages of becoming middle or upper class and explores the conflicts that changing one's social class can cause.
There are several conflicts worth mentioning in this play. First, Professor Higgins and Colonel Pickering have a playful "bet" that Higgins can't make a common flower girl speak and act like a duchess. There is the issue of language, pronunciation, and what we actually hear and say as opposed to what we think we hear and say.
There is the issue that Eliza Doolittle does want to improve her situation in life, but is rather stuck in her position as flower girl since her income fluctuates and her education is limited.
There is the issue of what to do with Eliza once she learns to speak and act properly. She can't go back to selling flowers, yet she has no real place among the aristocratic society she has been trained to infiltrate.
There is the conflict with Higgins and everyone else--including his mother--since he considerably rude, late, and isn't really the ideal role model for Eliza's "proper" education.
There is also the minor conflict of Eliza's father, who has come into some money himself and is struggling about his impending marriage and adjusting to suddenly becoming "a gentleman".
There is also the conflict of Eliza's romantic feelings for Higgins and for Freddy whom she meets at one of the outings to test her ability to fool others into thinking that she is high society.