What is the inside of a nut called?
I've always heard the "meat" of the nut, or "nutmeat". Alternate terms include "kernel" or "seed" or, well, "nut". 

If you were to ask a botanist, the edible part is the embryo and the endosperm, though it varies depending on which type of seed you're referring to. Also, for a few seeds—not sure if any of them are called nuts—we eat the seed coat as well (e.g., beans).