Huge Power doubles the Attack stat of the Pokémon with this Ability.
Because it is an Attack modifier and not a stat stage change, it will double the user's effective Attack stat when using Foul Play or Body Press.
Generation IV
Huge Power also affects self-inflicted confusion damage.
Generation V onward
Huge Power no longer affects confusion damage.
Pokémon Legends: Z-A
Because Abilities are not present in Pokémon Legends: Z-A, Pokémon that have Huge Power as their standard Ability have, instead, their base Attack stat increased by 40% exclusively in this game:
MegaMawile has its Attack stat increased from 105 to 147.
MegaStarmie has its Attack stat increased from 100 to 140.
Huge Power has a one-in-three chance of boosting a physical move by 50% when such an attack is used. The chance of activating appears to be lower in Pokémon Super Mystery Dungeon.
In Pokémon UNITE, Huge Power is Azumarill's Ability. When the user hits only a single opposing Pokémon with a basic attack, a move, or its Unite Move, the hit is guaranteed to be a critical hit. However, when the user hits two or more opposing Pokémon at once with a basic attack, a move, or its Unite Move, the hits are never critical hits.
Sometimes the power of the Pokémon's physical moves goes up.
Trivia
Its Japanese name appears to be a pun, as ちからもち (chikaramochi) is readable as both "muscleman" or "strengthening mochi".
This could explain why rabbit-like Pokémon, such as Azumarill and Diggersby, as well as Starmie, a Pokémon associated with outer space, have this Ability, being a reference to the moon rabbit, which is commonly depicted preparing mochi in Japanese folklore.
It could also explain why Mega Mawile has this ability, as its apparent inspiration, the futakuchi-onna is commonly depicted eating mochi.