In the Pokémon series
In the Pokémon series, each Pokémon has one or two types, and each attack has one type. Whenever an attack hits a Pokémon, the damage dealt is altered by any weaknesses or resistances caused by the type combination. For example, a Water attack will deal double damage to a Fire or Rock Pokémon, while a Grass attack will deal half damage to a Fire-type, and an Electric attack will have no effect on a Ground Pokémon.
In the Super Smash Bros. series
While type effectiveness is generally nonexistent in the Super Smash Bros. series, it does exist in a small capacity in Brawl. Pokémon Trainer's three Pokémon (Squirtle, Ivysaur, and Charizard) are affected by a stripped-down version of type effectiveness, but none of the other Pokémon characters are. Out of Fire, Water, and Grass attacks, if one of the three Pokémon is hit by an attack they resist, they take 10% less knockback. However, if the attack is super effective, they take 10% more knockback.
While the effect of type effectiveness is minor in Brawl, it is not insignificant. Most importantly, it causes Ivysaur to be KO'd at lower percentages by fire attacks than other characters, which is a problem compared to its two partners, since only Squirtle, Mario, Olimar, and Kirby (having copied Water Gun) have water attacks, and only Ivysaur and Kirby (having copied Bullet Seed) have grass attacks, but 19 characters can use fire.
The chain of effectiveness goes: Fire --> Grass --> Water --> Fire
In a similar way, Fire Primids appear to take no damage from fire moves. Moves that aren't purely fire (such as Falcon Punch) will still deal damage.