From the Super Mario Wiki
Ostros, also known as Ostriches, are large, flightless birds that first appeared in Super Mario Bros. 2, where they were ridden by Shy Guys. They are the first species that allow Mario to ride, even though they are enemies. In the credits and the manual of Super Mario Bros. 2, Ostro's and Birdo's names were switched, causing some confusion; this mistake was fixed in the remake, Super Mario Advance.
Super Mario Bros. 2
Ostros made their debut in Super Mario Bros. 2, where they were ridden by red Shy Guys that try to steer it towards the player. If their rider was dislodged the Ostro will travel in a straight line. Players can jump on top of an Ostro and ride it across obstacles regardless of whether or not it already has a Shy Guy rider. Ostros can be picked up with very little effort, but they don't bounce when thrown.
While Ostros are black in the NES version of the game, subsequent remakes have depicted them as being pink. Like many other enemies from Super Mario Bros. 2, Ostros have yet to make an appearance in a newer game.
The Super Mario Bros. Super Show!
Several Ostros, though usually referred to as Ostriches, appeared in several episodes of The Super Mario Bros. Super Show!, where they were usually rode and used to haul motorless vehicles, much like horses. Due to their role, Ostros never had any speaking lines, and usually made honking noises. They first appeared in the episode "Butch Mario & the Luigi Kid" where several Ostros were used by Mario, Luigi, Snifits and several other characters as steeds to travel through the Wild West. A wild ostrich appeared in "Too Hot to Handle" on the island of Waki-Waki. Mario, Luigi, and Toad rode it to reach Princess Toadstool faster.
Super Mario Bros. movie
Ostros were also given homage in the Super Mario Bros. film, which featured a sign with the word "Ostro" on it; this sign is thought to be advertising a company or organization of some sort.