The Shoulder Press or simply “Press” is the foundation of these three movements.  The main reason for the three different movements is to lift more weight or in CrossFit terms, move larger loads over a long distance quicker.  With the press you are solely relying on your upper body to move the weight.  The Push Press uses a “hip drive” to help pop the weight off the shoulders, giving the lifter the ability to move more weight.  Finally, in the Push Jerk, the lifter use hip drive and then proceeds to drop under the bar to help get the body low enough to land with arms locked out overhead in a quarter-squat position.  Then the lifter can rely on the larger, stronger muscles of the legs to help stand the weight up while still balancing the weight overhead.