Yoshisaur Munchakoopas

Therapoda Yoshisaur Munchakoopas (more commonly called Yoshi for short) is a green Yoshi which befriends Mario and assists him as one of his closest allies.

Super Mario World 2: Yoshi's Island
He finds Baby Mario who was dropped onto Yoshi's Island on accident when the Stork got attacked by Kamek. Yoshi helped Mario rescue his brother Luigi and defeated Kamek and his master the young baby Bowser.

Super Mario World
Years later, Bowser's Koopa Troop invaded all of Dinosaur Land and had all the Yoshi's trapped in their eggs. Mario who is now an adult rescued Yoshi and assisted him on his journey to save Dinosaur Land and Princess Peach as his mount.

Super Mario 64
While Yoshi does appear in Mario 64, he is not a playable character. After collecting 120 power stars, a cannon will open up to take you to the top of Peach's castle. On the top of the castle, Mario will find Yoshi who will give Mario a little message and 100 1-ups.

New Super Mario Bros Wii.

Yoshi will appear in some levels to help Mario clear them. Yoshi only appears in worlds 1-3, 2-5, 3-2, 4-5, 6-4, and 7-6.

New Super Mario Bros U.

While Adult Yoshies are not in the game, Baby Yoshies help Mario.

Super Mario Galaxy

While Yoshi doesn't appear in Super Mario Galaxy, there is a Planet shaped like his head.

Super Mario Galaxy 2

Yoshi assists Mario in some galaxies

Thanks to MatPat fore discovering Yoshi's scientific name