The crane is the easiest thing in the world and the hardest thing to explain. Drag the slider and the bird comes apart into the flat square it started as, one crease at a time.
The crane is not the easiest model in the world; the cup is. But the crane is the first one that makes somebody look up from the table, because it stands up on its own and has a head and a tail, and you made that happen with your hands and no glue.
The sequence of folds is also a proof. Each step is a transformation of the previous one, and the reason it works is that the paper has memory: every crease you make is a hinge that a later fold can use. The square at the end of this page is the same square you started with, with thirty hinges in it.