Effects are a property of an attack or hitbox. They appear in all 5 installments of the Super Smash Bros. series. While most of the effects only change the appearance of an attack, a few of them affect the gameplay.
Characters hit by attacks that have the fire effect will briefly ignite and flash orange. The effect is also capable of thawing frozen characters and detonating Blast Boxes in a single hit. In Super Smash Bros. Brawl, attacks with the fire effect does 10% more damage and knockback to Ivysaur, and 10% less damage and knockback to Squirtle. Red Pikmin are unaffected by this effect.
The electricity effect causes yellow and blue sparks to appear for a second around any opponent hit by an attack with the effect. It causes 50% more freeze frames than an attack without the effect would and slightly increases the hitstun. Yellow Pikmin are unaffected by this effect.
The water effect appears as a small splash where an attack with the effect hit the character. In Brawl, water does 10% more knockback and damage to Charizard, and 10% less knockback and damage to Ivysaur. Blue Pikmin are unaffected by this effect.
Grass is the rarest effect the Super Smash Bros. series. Ivysaur is the only playable character that has attacks with the effect. The Grass effect has an identical appearance and sound to the slash effect. It only affects the gameplay in Brawl where it does 10% more knockback and damage to Squirtle, and 10% less knockback and damage to Charizard.
The appearance of the slash effect is similar to a normal hit but with a white sideways arc. The effect is usually caused by swords or sharp weapons. It does not affect gameplay in any way.
Darkness is completely for appearance and has no effect on gameplay. Characters hit by an attack with darkness will emit a purple smoke and flash purple for a second. Ganondorf, Olimar, Mewtwo, and Robin are characters capable of using attacks with darkness.
The aura effect is unique to Lucario and only affects the appearance of an attack. When an attack with the aura effect hits a character, they will emit a blue smoke similar to the aura on Lucario's paws.
The PSI/Magic effect appears as a small explosion of sparkles where the opponent is hit. As its name suggests, it is usually found in magical or PSI infused attacks.
The Bury effect was introduced in Melee and it puts opponents to a buried state. This effect applies to the Pitfall item and Donkey Kong's Side Special. It also applies to some custom special moves in Super Smash Bros. for Nintendo 3DS/Wii U.
The Stun effect stuns the hit opponent for a short amount of time. Oddly, Stun is easier to escape at high percents unlike the Sleep or Buried states which are harder to escape at high percents. Mewtwo, Luigi, and Wario are the only playable characters that have a move with this effect.
Paralysis briefly immobilizes opponents allowing for a quick follow up. The duration of the paralysis varies depending on which attack is used and how long it is charged. Zero Suit Samus and Corrin are the only characters that can use paralysis attacks without custom moves.
The Freezing effect encases characters in ice, preventing them from moving for a short time. When hit with a move that freezes, characters are often launched purely vertically a short distance in addition to being frozen. The Freezie item is capable of freezing fighters it is thrown at; characters with moves that freeze include Lucas, Mega Man, Mr. Game & Watch, Kirby, Ness, and Luigi.