From Bulbapedia, the community-driven Pokémon encyclopedia.
The category (Japanese: 分類 category) of a move is an element of a move that determines the kind of damage it deals.
In games prior to Generation IV, the type of a damaging move determines whether the move is physical (damage calculation is performed using the user's Attack and the target's Defense) or special (the user's Special Attack and the target's Special Defense).
It was first assigned to individual moves in Pokémon XD for shadow moves, and then applied to all moves starting in Generation IV. Among fans, this change is often known as the Physical/Special split.
The Physical/Special split allows Pokémon to use moves from a wider variety of types. For example, prior to Generation IV, Sneasel, which has a much greater Attack than Special Attack, was not effective at using moves with STAB, due to Ice-type moves and Dark-type moves both being special.
Several new moves were introduced to give more balance to formerly all-special and all-physical types, such as the physical Psychic-type moves Zen Headbutt and Psycho Cut; and the special Fighting-type moves Focus Blast, Vacuum Wave, and Aura Sphere. Other moves, such as Hyper Beam (which was physical prior to Generation IV, but is now special), gained physical counterparts with the same characteristics (in this case, Giga Impact).
Prior to Generation IV, damage categories were as follows:
Shadow moves have damage categories assigned individually.