The inner square's area is exactly half the outer square's area.
The proof is really simple. Say the square has side of length s. Then, the radius of the inscribed circle is s/2. In the inscribed square, then, a segment drawn from the center of the square to a corner is s/2, and since that's the hypotenuse of a 45-45-90 right triangle, the apothem (half the length of the side of the little square) must be
So, the length of the side is , and you can see where I'm going. The big square's area was , and the little one's is , exactly half.
For the other case, a circle inside a square inside a circle, the relationship is exactly the same: The inner circle's area is half the outer one's!