An enemy that first appeared in Super Mario Bros. 3. A Chain Chomp is a vicious looking ball with sharp teeth, attached to a block by a chain. As Mario passed, Chain Chomp would quickly lunge in Mario's direction, making it an almost unavoidable enemy. The only ways to defeat it are to put on a Hammer Bros. suit and throw a hammer at it, or to put on a Tanooki suit, become a statue in midair, and land on the Chomp.
In Super Mario World 2: Yoshi's Island, Chain Chomps would usually wait on a faraway hill until Yoshi & Baby Mario passed, then fly up and come crashing down, creating a big gap in the path. In later levels, a giant Chain Chomp would chase after Yoshi, eating the soft land underneath Yoshi's feet. If Yoshi managed to outrun it, it would accidentally bite a hard block, stop just short of Yoshi, and fall down. A much smaller Chain Chomp resides chained to a block in Sluggy the Unshaven's Fort (World 5, Level 4). This particular chomp, when turned into a "?" cloud, would lead to a room with an information block that would tell you a secret code to let you play the Yoshi vs. Bandit mini-games. Also in the game was a round boulder called a Chomp Rock, which could roll over enemies and serve as a boost for Yoshi.
A giant barking Chomp appeared in Super Mario 64, chained to a post, guarding a Power Star on Bob-Omb Battlefield. Mario would have to Ground Pound the post 3 times (collecting a Red Coin in the process) to release Chain Chomp, who would jump off into the distance.
A few Chain Chomps appeared in Super Mario Sunshine as red hot puppies who left behind a trail of lava in Pianta Village. Mario would have to put them out with FLUDD (his combination water squirter/jetpack), pull their tails, and launch them into a pond. In a later episode, Mario faced a much larger Chomp, cooled it down, and dragged it to a spa to give it a hot bath.
The character originated from a bad experience Mario/Zelda creator Shigeru Miyamoto had with a dog as a child. Chain Chomps have also appeared in one of his other games, The Legend of Zelda: Link's Awakening, alongside cameo appearances from such Mario characters as Goomba, Boo, and Yoshi.
Zelda info provided by yerricde and Walter. Extra thanks to Servo5678