From the Super Mario Wiki
Grand Goombas (sometimes known as Giant Goombas) are the bigger counterpart of Goombas that have appeared as semi-recurring enemies throughout the Mario series.
 Super Mario series
 Super Mario Bros. 3
Grand Goombas make their first appearance in Super Mario Bros. 3, where they are only encountered in Big Island, with Gargantua Koopa Troopas and the Piranhacus Giganticus. Despite their massive size, they do not pose much of a threat. Jumping on them once or swinging Mario's tail with a Raccoon Suit or Tanooki Suit defeats them like regular Goombas.
 Super Mario 64 / Super Mario 64 DS
They appear again in Super Mario 64 and its remake in Tiny-Huge Island. If the player enters or warps to the huge side, the Goombas become Grand Goombas. These Goombas can only be defeated if jumped on, because punches and kicks don't affect them at all. Ground Pounding on them will produce a Blue Coin. Simply jumping on them also defeats them, but only produces a single yellow coin.
 Super Mario Galaxy
In Super Mario Galaxy, only one Grand Goomba appears in the entire game. It appears in the Gateway Galaxy, and must be defeated to get a key in order to both free a trapped Luma and head inside the planet to retrieve a Grand Star. The Goomba only appears when the player speaks to the trapped Luma. Grand Goombas appear much larger then they did previously. They also make a loud pounding noise when they chase Mario. Spinning them or hitting the nearby metronome puts them on their head, and a kick while they're stunned takes them out.
 New Super Mario Bros. Wii
Grand Goombas reappear in the game New Super Mario Bros. Wii. Here, when jumped on they split into two normal Goombas. A larger species called Mega Goombas also appear and are two times larger than a Grand Goomba. When a Mega Goomba is jumped on, it splits into two Grand Goombas. They appear in the World 1 Enemy Course and in World 8-2.
 Super Mario Galaxy 2
In Super Mario Galaxy 2, Grand Goombas return as enemies. They only appear in the Supermassive Galaxy. Like in the predecessor, Mario can attack them by spinning them and then kicking them to defeat them. Paragoombas also get their own giant counterpart in this game, called Huge Paragoombas.
 Super Mario 3D Land
Despite Grand Goombas not appearing in Super Mario 3D Land, a subspecies called Grand Tail Goombas do; they are Grand Goombas that have Tanooki tails. They appear only rarely, and have the same attack pattern as a normal Tail Goomba. As such, one stomp to the head or swipe of the tail with a Tanooki Suit defeats one.
 New Super Mario Bros. U
Grand Goombas make an appearance in New Super Mario Bros. U. In this game, they only appear in the level Jungle of the Giants in Soda Jungle, appearing when the player jumps on a Mega Goomba. Their behavior is the same as in New Super Mario Bros. Wii. The Prima Guide referred to them as "Hefty Goombas," which is false.
 The Adventures of Super Mario Bros. 3
In the episode "Princess Toadstool for President" Bowser and his son Cheatsy had a plague of Huge Paragoombas in a cage then release them to eat all of Grass Land, but when Bowser push a cage at Mario and Luigi which they leap on it then they stomp on the two making them Grand Goomba, but the two Grand Goomba try to attack them but slam into each other. Their was an error on this episode where two more Huge Paragoomba try to attack them but slam into each other but they are seen as Grand Goomba but they suddenly disappear.
 Nintendo Adventure Books
An uncommonly large Grand Goomba, thought to be the progenitor of the sub-species, appears in Koopa Capers, having been presumably brought to Dark Land by Wendy O. Koopa, who needed some of its leg hairs for a potion she wants to create. After crossing a Podoboo-infested body of lava in the Magma Pits, Luigi encounters the Grand Goomba wandering around aimlessly, and if he decides to crawl between its legs, he will go unnoticed by the giant fungus. If he decides to try and jump onto a nearby ledge instead however, the Grand Goomba will grab him mid-leap, stuff him in a sack, and begin to head home to the Land of the Giants, muttering that Luigi will make "a lovely plumber omelet".
 Super Smash Bros. Brawl
The Grand Goomba (referred to as a Giant Goomba in the trophy description) later appeared in Super Smash Bros. Brawl as an enemy in the Subspace Emissary that takes a few more hits to defeat than a regular Goomba. It is not to be confused with the Mega Goomba from New Super Mario Bros..
 Mario Kart 7
Grand Goombas appear in Mario Kart 7 for the Nintendo 3DS. They act as obstacles to racers and cause a kart to spin-out if hit. They appear in Mario Circuit, Piranha Plant Slide, Wii Mushroom Gorge and Wii Koopa Cape. They replace Goombas which appeared in past installments.
 Trophy Information from Super Smash Bros. Brawl
 Names in Other Languages