Someone who's loony is nutty or wacky — it's a slang term for "crazy." Your loony aunt Mabel might prefer wearing a teddy bear costume when she goes to the grocery store.
Loony is an informal, derogatory term for mentally ill, but it can also mean silly or outrageous. You could say, "My parents are loony — they call me three times a day and are always sending me photos of their little dog dressed in doll clothes," or talk about the loony story you read for your book group. Loony is an American-coined word, short for lunatic and also referencing the cry of the bird called a loon.