Damage categories are the classification of moves in the Pokémon series. There are 3 main types targeting different stats with exceptions. Moves also have some subcategories that affect certain things. Prior to the fourth generation, damage categories were locked to Pokémon types rather than an independent classification
Physical moves rely on the user's Attack stat against the opponent's Defense stat.
Special moves rely on the user's Special Attack stat against the opponent's Special Defense stat.
Status moves affect the stats, status condition and everything else. Damage inflicted through these is completely unaffected by either Pokémon stats aside from the target's HP.
Contact moves are moves that make physical contact with the opponent. These are mostly physical moves.
Sound moves are a group of moves that are ignored by Soundproof. They also have the property of ignoring Substitute. All these moves are Special moves or Status moves.
Punch moves are moves that are powered up by Iron Fist. They are all physical moves and most have the word punch in them