Botanically, a nut is a fruit whose pericarp (i.e., the entire ovary wall) forms a shell that doesn't split open protecting a single seed. This definition leaves out peanuts (their shells form only from the mesocarp, or middle layer of the ovary wall), but also almonds, pecans, cashews, macadamias, coconuts, and pistachios, all of which are botanically drupes, whose shells form from the endocarp (inner layer of the ovary wall) and which grow inside of a fleshy fruit. Basically, they're edible peach or cherry pits. Brazil nuts are also out, because they have 10-25 seeds per pod, as are pine nuts, because they come from pine cones, and walnuts, which are pseudo-drupes.

In short, if you're going to insist upon the strict botanical definition for nuts humans eat, you'll be left with chestnuts, hazelnuts, and not much else. The culinary definition (a dry, edible seed) is much more useful day to day.