From Bulbapedia, the community-driven Pokémon encyclopedia.
Damage categories are categories introduced in Pokémon XD for shadow moves, and then expanded to all existing moves in Generation IV, that categorize moves based on the kind of damage they deal. In games of previous Generations (other than Pokémon XD), a move would base its power on the user's Attack and the target's Defense or the user's Special Attack and the target's Special Defense depending on its type. For example, all Fighting-type moves were the former, while all Fire-type moves were the latter. However, in Generation IV, a move like Focus Blast is both Fighting-type and bases its damage on the Special Attack and Special Defense of the user and target, respectively.
The addition of this feature to the games has altered the playing field somewhat, as now Pokémon are allowed a much greater degree of freedom with the types of move that can be used. For example, under the former system, Sneasel, which has a much greater Attack than Special Attack, would often be forced to ignore all moves that would give it STAB, due to Ice-type and Dark-type moves both being calculated from its Special Attack, the base of which is a low 35, compared to its Attack base stat, 95.
Overall, when the feature was first announced, the change was very controversial among fans, as many players were unsure of which moves from the previous three generations would be affected. Eventually, this fear of losing hard-trained Pokémon due to the conversion died down, and as movesets were revealed, the change became an accepted part of the game.
While the moves remain, for the most part, in the same category that they were in before, the majority of moves of a formerly "Special" type that were given status as Physical, as well as moves of a formerly "Physical" type that were given status as Special, are obvious changes, as is the case with Fire Punch, Ice Punch, and Thunder Punch. Several new moves, as well, were introduced to give more balance to formerly all-Special and all-Physical types, such as the Psychic-type, which got Zen Headbutt and Psycho Cut, and the Fighting-type, which got Focus Blast, Vacuum Wave, and Aura Sphere.
Moves that deal no direct damage were also given a category; however, as they never dealt damage in the first place, no moves formerly of other categories were moved to this category, and no moves moved from this category to another.
Prior to Generation IV, damage categories were as follows: