What does this exchange about a handkerchief and a sleeve mean?

In My Fair Lady the following dialogue ocurs:

ELIZA: What's this for?

HIGGINS: To wipe your eyes. To wipe any part of your face that feels moist.

Remember, that's your handkerchief and that's your sleeve.

Don't confuse one with the other, if you want to become a lady in a shop.

What does it mean?

Best Answer

To summarise from Wikipedia, the story of My Fair Lady is as follows:

The story concerns Eliza Doolittle, a Cockney flower girl who takes speech lessons from professor Henry Higgins, a phoneticist, so that she may pass as a lady.

In other words, taking a common girl and making her into a proper lady.

The lines you refer to are the professor telling her that a proper lady would wipe her face with a handkerchief, not her sleeve. Only a common girl would use her sleeve. Thus, if she used her sleeve it would give away that she was from a common background - which would be bad, since Henry wants her to pass as a lady.