Anything you do vainly does not result in the outcome you're hoping for. When you're vainly trying to fall asleep, the night can seem to last far longer than eight hours.
If you fall vainly in love with someone, it's hopeless — it's never going to work out between the two of you. And if you try vainly to convince your friends not to gossip about a classmate, they just won't listen to you despite your efforts. You can also use this adverb to mean "in a conceited way," like when the evil queen in Sleeping Beauty stares vainly into her magic mirror asking, "Who's the fairest one of all?"