One of the Princesses of Heart, her world was destroyed by Maleficent.
Kingdom Hearts
She was taken and held deep within Hollow Bastion until all seven Princesses were collected. It is unknown when she was captured but from scenes it can be assumed it was before Snow White and Alice placing her as either 2nd captured or 3rd either before or after Belle. She and the other princesses upgraded Sora's Fire spell to Firaga. She first appeared in Hollow Bastion in Kingdom Hearts.
When Kingdom Hearts was sealed she was returned to her own world.
Kingdom Hearts II
Cinderella is absent in Kingdom Hearts II, though her name was part of the password that allowed access to the DTD dataspace in Tron's world.
Birth By Sleep
She will, however, return in the upcoming PSP game Kingdom Hearts: Birth by Sleep. The latest trailer implies it is Ven and Mickey who created her dress and somehow give it to Fairy Godmother to cast on Cinderella. However, Ven had an obstacle: Lady Tremaine's (Cinderella's wicked stepmother) equally evil cat Lucifer.
Abilities
Because Cinderella is a Princess of Heart, her heart holds part of the ability to open the Final Keyhole that leads to Kingdom Hearts. She also has the power to hold back darkness using the light stored within her heart, a power which she and the other princesses can utilize alone (the exact way she fights off darkness is currently unclear).
In addition, her heart (along with that of the other princesses) can be used to create the Dark Keyblade, which allows the bearer to unlock anyone's heart.