From the Super Mario Wiki
Bumpties are a race of penguin-like creatures that first appeared in Super Mario World 2: Yoshi's Island. They are usually found in icy regions. Most Bumpties are blue, however pink and green individuals were shown in Paper Mario. Their name is a corruption of "bumpy", referring to their bouncy, rubbery feathers.
Super Mario World 2: Yoshi's Island / Yoshi's Island: Super Mario Advance 3
Bumpties are usually found wandering around in icy areas of Yoshi's Island in their debut game, Super Mario World 2: Yoshi's Island. While Yoshi can grab Bumpties with his tongue, he cannot eat them. Also, if he runs into one, he is bumped backwards, hence the name. This means that Bumpties can't hurt Yoshi.
They can also steal Baby Mario if Baby Mario has already been knocked off Yoshi and is floating towards one, like a Bandit.
A certain Bumpty appears in Tetris Attack, allied with Dr. Freezegood. In here, he is depicted as Yoshi's friend, but enchanted by Bowser and his minions to work for him. Yoshi and Bumpty become friends again after freeing him and Dr. Freezegood from the spell. He also notes in the game, that others say that he's cute, but he really doesn't like it, and would like to be said as cool.
Note that Bumpty only appears in the SNES version of Tetris Attack. He does not appear in the Game Boy version.
Yoshi's Woolly World
Yoshi's Island DS
Yoshi's New Island
Paper Mario series
Bumpties in the Paper Mario series are referred to as penguins.
Bumpties return in Paper Mario. In this game, Mario travels to a city of Bumpties called Shiver City. He also meets a famous writer named Herringway who is a Bumpty. In this game all the Bumpties are allies, not enemies. A small group of Bumpties (Including Mayor Penguin) appear in Princess Peach's Castle prior to the arrival of Bowser.
Paper Mario: The Thousand-Year Door
Bumpties also make a small appearance in Paper Mario: The Thousand-Year Door where they live in Poshley Heights, a town of the rich and famous. Mario also meets an incompetent detective Bumpty named Pennington.
Super Mario series
Super Mario Advance 4: Super Mario Bros. 3
Bumpties also appear in Super Mario Advance 4: Super Mario Bros. 3, only appearing in the Japan-Only Level Tā-kun ni Ki o Tsukero. The sprites are exactly the same as in Yoshi's Island. Players can not harm them unless they use a Fire Flower or Hammer Suit. If players try stomping on it, they are bounced off, and if they touch it from the side, players will not be harmed, but they will be knocked off to the side.
Names in other languages