Jasmine

Jasmine is one of the Princesses of Heart who appered in all of the Kingdom Hearts games. She lives in the palace of Agrabah along with Genie and Aladdin.

Kingdom Hearts
In Kingdom Hearts she was hiding from Jafar. When Aladdin, Sora and Co. arrived Jafar disappeared with her to the Cave of Wonders. When Aladdin, Sora and Co. were fighting Jafar, Riku came and took her to Hollow Bastion.

Her heart was stolen and used to open the Final Keyhole. With the help of the other five princesses, she used her power to contain the darkness lurking out of the keyhole until Sora arrives and seals it.

When Kingdom Hearts was sealed she was returned to her own world.

Kingdom Hearts: Chain of Memories
Jasmine appeares as a fragment of Sora's memories. She is kiddnaped by Jafar and is later rescued by Sora.

Kingdom Hearts II
In Kingdom Hearts II she is first seen at the Palace. She noticed that Aladdin often seemed depressed and was worried about him. In the mean time, Jafar's parrot, Iago did his best to earn everyone's trust, and although Jasmine did not trust him at first, she eventually learned to do it.

When Sora and the others returned, they found out that Jafar had been released and went on a false lead to find him by Iago, and she was kidnapped by Jafar but Aladdin and Sora saved her in time.

Jasmine's name was found out to be part of the password that allowed access to the DTD dataspace in Tron's world.

Dreams
Jasmine has a longing for freedom and true love, much to her father's dismay.

Abilities
Because she is a Princess of Heart, she holds the power to open the Final Keyhole when she is joined with the other princesses.

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.

She also has the power to utilize the darkness with the aid of the other princesses.

Jasmine (along with the other princesses) can also upgrade other people's powers, as revealed when they upgraded Sora's Fira spell to Firaga.

Trivia

 * A pillar in the Awakening depicts three silhouettes resembling another Princess of Heart, Alice. This hints on the fact that at the moment of the Awakening, three Princesses of Heart (one of which is Jasmine) have yet to be captured.