From the Super Mario Wiki
A Fireball (also written as Fire Ball) is a magic ball of fire that can be used as a weapon. Typically, one can only shoot fireballs if he or she acquires a Fire Flower. Some enemies can shoot fireballs without the use of a power-up. (Additionally, Mario and Luigi can shoot fireballs from their hand without using a Fire Flower). Ice Balls can be seen as a parallel to Fireballs.
Super Mario series
Super Mario Bros.
Super Mario Bros. 3
Fireballs return in Super Mario Bros. 3, with being used as a weapon again. Though Mario and Luigi aren't the only ones that can shoot fireballs, as Fire Bro's can also shoot fireballs from their mouths which acts like the Mario Bros. fireballs.
Super Mario World
Fireballs appear in Super Mario World where they are used as weapon. Additionally, Yoshi can breathe fireballs if he eats a Red Shell and Red Yoshi can also breathe fireballs if he eats a Koopa Shell. The fireballs that Yoshi spit out look like Podoboos and destroy most enemies.
Super Mario 64 DS
New Super Mario Bros. Wii
Fireballs appear in New Super Mario Bros. Wii. Like in previous games, they are used as weapon. Yoshi can also spit a giant Fireball by eating a fireball launched by a character with a Fire Flower power up, or from a Fire Piranha Plant. This giant fireball transforms all enemies encountered into coins.
Mario Kart series
Super Mario Kart
Fireballs make their debut in the Mario Kart series in Super Mario Kart, where Bowser would drop these items and touching them would make the driver spin out. Unlike other laid down items such as Banana Peels, however, they would move around a bit on the course.
Mario Kart: Double Dash!!
Fireballs reappeared in Mario Kart: Double Dash!!, where they are the special item of Mario (and Petey Piranha and King Boo, due to their ability to have any special item). Luigi's is a Green Fireball, but it acts the same. When the special item is used, five fireballs will be launched into the racetrack, hitting other drivers and causing their karts to spin-out. These fireballs are capable of ricocheting off walls. However, after a short amount of time, the launched fireballs will fizzle out.
Mario Kart Wii
Fireballs also appear in Mario Kart Wii, where they only appear in the 2nd Competition of March 2009, with Dry Bowser spiting out fireballs to attack the player. He'd either fire three fireballs backwards or one fireball forward, depending on the player's racing position.
Mario Kart 7
Mario Kart 8
Fireballs appeared in Mario Kart 8 where they re-use their role from Mario Kart 7.
Mario Superstar Baseball
Fireballs also appeared in Mario Superstar Baseball, where they are Mario and Luigi's special pitch. It, unlike most other pitches, goes in a straight line. The ball, as the name says, turns into an extremely hard-to-hit fireball (red for Mario, green for Luigi).
Mario Super Sluggers
Fireballs reappeared in Mario Super Sluggers. Like in Mario Superstar Baseball, it is used as Mario's Star Pitch, though Luigi is given a different Star Pitch. Just like in the predecessor, Mario will throw the baseball at a fast speed, which makes it harder for batters to properly hit it.
Club Nintendo Comics
Super Smash Bros. series
Fireballs appear as the neutral special move used by both Mario and Luigi throughout the Super Smash Bros. series. Mario's Fireballs are affected by gravity (meaning they bounce against the ground), while Luigi's shoot straight no matter what, unless it ricochets off of something in front of Luigi. Kirby can also use the Fireball move once he has sucked up Mario or Luigi, and the color and performance of the Fireball are according to which Mario Brother he has sucked up. In Super Smash Bros. Brawl and Super Smash Bros. for Nintendo 3DS / Wii U, Yoshi can spit single fireballs in his Final Smash form.
In Super Smash Bros. for Nintendo 3DS / Wii U, the first custom variant of Mario's Fireball is Fast Fireball, in which the fireball travels directly forward at high speed but doesn't cause knockback and deals less damage. The second is Fire Orb, which fires a larger but slower fireball that moves similarly to the standard Fireball but can hit multiple times. The custom variants for Luigi's Fireball are Bouncing Fireball, which behaves like Mario's standard Fireball, and Ice Ball, which moves more slowly but can freeze opponents.
Names in other languages