If someone's batty, she's eccentric or a little bit wacky. You might love it when your batty great aunt visits because she makes life so interesting.
The adjective batty can be derogatory — as when you dismiss a mentally ill person as batty — or fond, as when you praise your batty English teacher, who quotes Shakespeare while doing handstands. If someone says, "You are driving me batty," it means you're making them feel crazy. Batty originally meant "like a bat," with the "eccentric" meaning arising from the expression "bats in the belfry," which means "crazy" or "not right in the head."