The center of mass of objects plays an important role in the way things will move through space. Gravity is a constant force of nature where there is always a struggle to achieve a balance with it. The center of mass affects not only the path that it will follow, but also affects how it will behave to counteract the force acting upon it.
When objects are thrown into the air, the path that they travel are in arcs or in a parabola. Because they are in motion, there is also a tendency for the object to rotate on its own axis which goes through its center of mass or gravity.
An example is a ball thrown in the air, where it will not only travel in an arc, but it will also spin around on its own axis as it travels in that arc. In this case, the center of mass would be the center of the ball because the weight is distributed evenly. In another example, such as a hammer, the weight is not distributed evenly because it has a handle. In this case it would rotate about the center of mass of that object, which would be most likely in the top portion of the handle just below the hammer's head.
When objects are thrown into the air there is also some deformation going on with that object in motion. Because gravity is acting upon it the object will go through some "squash and stretch".
