A nut is a kind of fruit made of a large, edible seed with a very hard shell around it. The nuts that squirrels typically gather (sometimes storing them temporarily in their cheek pouches) are acorns.
It might seem strange that a nut is a fruit, since until they've been cracked open, most nuts are as hard and inedible as rocks. Think of the meat of a walnut or almond that hides inside — this is the edible seed within the fruit. If someone calls you a nut, or describes you as nuts or nutty, they think you're crazy or wacky, and when you refer to "a tough nut to crack," you're talking about a tricky problem.