From the Super Mario Wiki
A Green Fireball is Luigi's trademark weapon in most Spin-Off games. It is identical to Mario's trademarked Fireball, except the fireballs are green instead of the usual red. Although Luigi has used this in many games, it seems that the ability might be replaced by electricty, or plasma, as seen in Mario & Luigi: Superstar Saga and Mario Strikers Charged.
Super Smash Bros. series
Green Fireballs made their debut in the Super Smash Bros. series as Luigi's special move. Although the move was called Fireball, the same name as Mario's move, it differed slightly in both appearance and effect. Luigi's fireballs were green, matching his choice of clothing, and the fireballs were not effected by gravity. This means the fireball will always go straight, and won't bounce on the ground. It also does slightly more damage. However, other than that, nothing else was changed. Kirby can use Green Fireballs when he sucks up Luigi.
In Super Smash Bros. for Nintendo 3DS / Wii U, the green fireballs can be altered into either Fast Fireball or Ice Ball, provided that they are unlocked first. Fast Fireball behaves like Mario's standard Fireball, while Ice Ball has a chance of freezing opponents.
Mario Kart: Double Dash!!
Green Fireballs appeared in Mario Kart: Double Dash!!, as Luigi's special item, and can only be used by him. It works the exact same way as Mario's Fireball item. When Luigi uses the item, either forward or backwards, he will throw five green fireballs across the course, which will ricochet off walls. It will burn any player it hits, which will daze them for a small while. However, the move can ricochet back at Luigi, dazing him as well. The fireballs split apart the farther they go, making it less accurate, so it is best used when close up.
This is not to be confused with the Green Fire, Luigi's kart in the same game.
Mario Superstar Baseball
A Green Fireball appeared in Mario Superstar Baseball, as Luigi's Power Shot. When used while Pitching, he will cover the ball in green fire, and toss it towards the batter at incredible speeds. This makes it hard to hit the ball, as it can easily zoom right past the batter. When used while Batting, Luigi will cover it with green fire when he hits it, and it will zoom past the field at incredible speeds. If someone catches it before it lands, they will catch fire and drop the ball. If it lands, it will lose its fire, and can be picked up safely. In both ways of using the move, it works exactly the same as Mario's Fireball special move. In Challenge Mode, Luigi will only be able to use this when the item, aptly named Green Fireball, has been bought.
Names in other languages
Mario Hoops 3-on-3
A Green Fireball is used in Mario Hoops 3-on-3, as Luigi's Special Shot. When the player taps out the letter L on the touch screen, Luigi will start charging the ball with green fire. If he is not hit while charging it, he will leap up into the air, hold the now fiery ball in his hand, and launch it to the basket. It will always get in the basket if it is shot inside the 3-Point line. When shot outside the 3-Point line, he will throw it, but someone else has to dunk it for him. The move is an exact copy of Mario's Fire Shot.