In the past, the distinction between lady and woman used to be clearer:
woman = general word for a female adult
man = general word for a male adult
lady = an honorable woman with good manners and refined behavior
gentleman = an honorable man with good manners and refined behavior
Nowadays, however, there is not so much of a difference. Although the words lady and gentleman still carry some connotation of being refined and well-mannered, people often use the words even to describe people they don't know.
An old lady fainted at the train station and they had to call an ambulance.
Good evening, gentlemen. Can I get you anything to drink?
The word girl is usually only used for female children or adolescents. Adult women sometimes refer to their friends as girls, but it can be considered offensive for an adult man to refer to adult women as "girls."