From the Super Mario Wiki
Dolphins are goggle-wearing cetaceans (marine mammals) that jump out of the water in Super Mario World.
Super Mario World
Dolphins only appear in Vanilla Secret 3 and in a hidden area near the end of Chocolate Island 1. They are used as platforms to cross the water infested with Porcupuffers. They jump in and out of water either vertically or in parabolic arcs. They appear on the "enemy list", but are described as "not really an enemy", though it is said "they can take Mario the wrong way". In the Japanese version of Super Mario World, Yoshi can eat the Dolphins. This was changed in the Western versions and later versions like Super Mario Advance 2 for the Game Boy Advance.
Mario Party series
Mario Party 4
A Dolphin appears in Koopa's Seaside Soiree in Mario Party 4, where he will let the character played as get on his back while he swims through the water. The character can Jump to get Coins this Dolphin swims by on the trip.
Mario Party Advance
An unnamed, comedian Dolphin is met in Mario Party Advance; occasionally, several Dolphins will appear and form an audience to listen to this comedian Dolphin's jokes.
Mario Party 5
Mario Party 7
In Mario Party 7, some Dolphins can be seen jumping out of the water in the file select screen.
Mario Party 8
In Mario Party 8, the Dolphins play an important role in the board Goomba's Booty Boardwalk; players can pay them a certain number of coins depending on what place they are in, allowing them to take a ride on the Dolphin's back and advance a number of spaces.
Mario Party 9
In Mario Party 9, Dolphins appear on Blooper Beach. On the first half of the board, a Dolphin is the first active fish when a party is started, being six spaces away at first. They advance two spaces every turn, and if the player lands on or catches up to the dolphin while captain, the Dolphin gives them 5 Mini Stars. Dolphins also appear when the player lands on a Lucky Space near the end of the board. Landing on one of them results in a trio of Dolphins leading the player to an island with several Mini Stars. Additionally, during the minigame Ring Leader, the players ride Dolphins to jump through hoops.
Mario Kart series
Mario Kart Wii
Mario Kart Arcade GP DX
Mario Kart 8
In Mario Kart 8, there is a banner with Dolphin artwork from Super Mario World on it found in Toad Harbor. Also, Dolphin Shoals features Dolphins that act similarly to those in Mario Kart Arcade GP DX, as well as Dolphin-shaped hill structures. In Sunshine Airport, this course can be seen listed on the signboards in the terminal next to other courses.
Mario Golf: World Tour
Super Mario World television series
In the Super Mario World television series, Dolphins appear only in the episode "Mama Luigi". In this episode, two Dolphins, seeing that Luigi and Yoshi are being attacked by a Torpedo Ted, decide to help the two by letting them ride on their backs. Evading the Torpedo Ted, the Dolphins drop Luigi and Yoshi off at a nearby beach. After being thanked by Luigi, the two friendly Dolphins leap back into the water, cheering happily.
Nintendo Adventure Books
A Dolphin is featured on the cover of Dinosaur Dilemma, and artwork of one is used in the book, but none appear in the actual story. An equally friendly sub-species of them, the Micro-Dolphins, are discovered to be living inside of Yoshi in Unjust Desserts.
Names in other languages