Contact or direct attack is a term used to describe the condition when one Pokémon in a battle physically touches the opponent in order to deal damage with a move.
Contact is distinct from whether a move is physical or special in nature, but most moves that make contact are physical moves, while most special moves do not make contact; status moves do not make contact in any case.
The use of contact-based moves can trigger various Abilities of the foe, such as Cute Charm and Effect Spore. It can also trigger certain items, and can be used to spread Pokérus.
Hitting a substitute does not make contact with the Pokémon behind it.
These are physical moves that do not make contact. (This does not include moves such as Hyper Beam, which belonged to physical-based types before Generation IV reclassified them as special.)
Also, while confusion damage is physical, it does not make contact.
List of special moves that make contact
These are special moves that do make contact. (This does not include moves such as Fire Punch, which belonged to special-based types before Generation IV reclassified them as physical.)
Each hit from a contact move from a Pokémon with this Ability has a 30% chance (20% chance in the Japanese versions of Black and White) of poisoning the target.
Damages the holder each turn for 1/8 of their max HP. Transfers to an attacker on contact.
Trivia
In Pokémon Colosseum and XD: Gale of Darkness, battle animations for contact moves do not show the attacking Pokémon hitting the defending one at all. As in the Stadium series, physical blows usually utilize quick camera cuts between the attacking and defending Pokémon to imply direct contact.