Five Abilities are also adapted especially for Double Battles. These
Abilities are:
Plus - If a Pokémon with Plus is in battle on the same side of the
field as a Pokémon with Minus, its Special Attack will be boosted by
50%. In Generation V and beyond, the effect is also present if another
Pokémon on the same side has Plus.
Minus - If a Pokémon with Minus is in battle on the same side of the
field as a Pokémon with Plus, its Special Attack will be boosted by
50%. In Generation V and beyond, the effect is also present if another
Pokémon on the same side has Minus.
Lightningrod - If a single-target Electric-type move is used, it will
be forced to strike the Pokémon with this Ability, regardless of the
Pokémon originally selected as the target and regardless of the move's
accuracy. Moves which target multiple Pokémon cannot be redirected. In
Generation V and beyond, the move will raise the Special Attack of the
Pokémon and deal no damage to it, unless the Pokémon is immune to the
attack by nature of being a Ground-type.
Storm Drain - If a single-target Water-type move is used, it will be
forced to strike the Pokémon with this Ability, regardless of the
Pokémon originally selected as the target and regardless of the move's
accuracy. Moves which target multiple Pokémon cannot be redirected. In
Generation V and beyond, the move will raise the Special Attack of the
Pokémon and deal no damage to it.
Telepathy - A Pokémon with this Ability will avoid damage from any
moves used by its allies, whether they directly target it or target it
as well as opponents.
That all came from here. However, this is only 5th generation, 6th gen introduced 3 more doubles-specific abilities, and 1 that works with allies:
Flower Veil: Prevents lowering of ally Grass-type Pokemons stats
Symbiosis: This Pokemon passes its item to an ally after their item is
consumed.
Healer: 30% chance of curing an adjacent ally's status at the end of
each turn.
Aroma Veil: Protects allies from attacks that limit their move
choices.
That all came from Pokemon Showdowns teambuilder.
And now for moves...
Several different moves are made specifically for Double Battles, such
as Helping Hand. Some others, such as Surf, can have noticeably
different effects. Moves that hit multiple Pokémon have their damage
reduced by 25%, unless all other Pokémon have fainted at that point.
The moves After You, Quash, Helping Hand, Ally Switch, Follow Me, Rage
Powder, and Aromatic Mist will have no effect if not used in a Double
or Triple Battle.
The move Flame Burst takes away 1/16th of the maximum HP of the other
Pokémon on the same team as the target Pokémon. This damage is not
treated as an attack.
The move Acupressure, while in a Double Battle, can target the ally
instead of the user.
That came from here, which also provides a comprehensive chart to show which moves will hit what in doubles.