Using Hubble Space Telescope images of 119 young planetary nebulae (PNs), most of which have not previously been published, we have devised a comprehensive morphological classification system for these objects. This system generalizes a recently devised system for pre-planetary nebulae, which are the immediate progenitors of PNs. Unlike previous classification studies, we have focused primarily on young PNs rather than all PNs, because the former best show the influences or symmetries imposed on them by the dominant physical processes operating at the first and primary stage of the shaping process. Older PNs develop instabilities, interact with the ambient interstellar medium, and are subject to the passage of photoionization fronts, all of which obscure the underlying symmetries and geometries imposed early on. Our classification system is designed to suffer minimal prejudice regarding the underlying physical causes of the different shapes and structures seen in our PN sample, however, in many cases, physical causes are readily suggested by the geometry, along with the kinematics that have been measured in some systems. Secondary characteristics in our system, such as ansae, indicate the impact of a jet upon a slower-moving, prior wind; a waist is the signature of a strong equatorial concentration of matter, whether it be outflowing or in a bound Keplerian disk, and point symmetry indicates a secular trend, presumably precession, in the orientation of the central driver of a rapid, collimated outflow.