I have included some more links below to the enotes study guide section on this play to give you more information on the necessarily brief summary that I will have to give you in response to this question. The play concerns one man's inability to change one human being into a very different kind of human being, as the mythological allusion makes clear. Pygmalion was a stonecarver who created a statue of a woman that he fell in love with. The Gods took pity on him and gave the statue life, and Pygmalion married this woman.
In Shaw's satirical equivalent, the God who is doing the changing is a man called Henry Higgins, who is so obsessed with accents and dialects that he believes he is able to transform Eliza Doolittle, a cockney flower seller, into somebody that would be perfectly at home with dukes and duchesses by changing the way that she speaks. Henry Higgins therefore embarks on his project of transforming Eliza Doolittle, which he does with great success. The only problem is, he has to decide what to do with her now that she has tasted high society and will not retreat easily into her working class obscure origins once more.