If planet orbits are in the shape of an ellipse and the sun is at one focus, what is at the other focus? I've studied ellipses. I've studied physics. But when it comes down to the elliptical orbits of the planets is where I get confused.
Ellipses contain two foci — and in the orbits of our solar system the sun is consistently stated as one of them. Okay. What about the second focus point? Why is it never brought up? And if there is no second focus — then how do we even move in an elliptical orbit?
 A: There is no physical object at the location of the second focus.  
Newton showed that an elliptical path was the consequence of an inverse square radial force from a fixed point.  While you can identify the point that is the second focus, nothing associated with that point is required to create the elliptical motion.
Deriving Kepler's Laws from Inverse-Square Law
A: Due to the force of gravity, which goes as the inverse of the square, planets trace out an ellipse in space as they orbit around the sun, which is located at a single focus. The other focus is unphysical. 
Actually, given two massive bodies, their "difference" vector will trace out an ellipse with the center of mass at the focus. Because the sun is so much more massive than most of the planets, we can usually disregard this since the center of mass is located within the sun itself.
Without going into more complicated effects such as orbit precession, here is an image depicting a planet's motion about an approximately stationary sun:

