A Grand Goomba is a large version of a Goomba.
Super Mario Bros. 3
They can be found in Giant Land in the game Super Mario Bros. 3. However, despite these Goombas being several times larger than a regular Goombas, they pose no more of a threat; stomping on them like you would do with a regular Goomba does the same job.
Super Mario 64 series
They re-appeared in the game Super Mario 64 and its Nintendo DS remake Super Mario 64 DS in Tiny-Huge Island. When the player entered the huge side, the Goombas would enlarge into Grand Goombas. These Grand Goombas can only be defeated by jumping on them. Ground Pounding them will uncover a Blue Coin.
In Super Mario Galaxy, a Grand Goomba is an enemy that must be defeated in the Gateway Galaxy in order to get the key that will allow Mario into the planet it lives on. In Super Mario Galaxy 2, they appear in Supermassive Galaxy, accompanied with the Grand Paragoomba.
In Super Smash Bros. Brawl, they re-appear under "Giant Goomba" as an enemy from the Subspace Emissary. They will take more abuse than a regular Goomba before calling it a day. It is not to be confused with the Mega Goomba from New Super Mario Bros..
They appear again in Super Smash Bros. for Nintendo 3DS in Smash Run mode. They are just like their smaller counterparts and attack the same way, only stronger. They can be defeated by either attacking normally or stomping three times on their heads.
Speaking of which, both the Mega Goomba and the Grand Goomba re-appear in New Super Mario Bros. Wii. Here, when the Grand Goomba is jumped on, they will split into two single Goombas; when a Mega Goomba is jumped on, they will split into two Grand Goombas.