Another great image made by jadenkaiba, in which you can see each one of the forms Kairi can get during the story of The Dark Lord/The Phoenix Strike.

As they're not in order of acquisition, first I will name each Form in the order from left to right:

1-Titan Form
2-Ramuh Form
3-Ifrit Form
4-Diabolos Form
5-Esper/Phoenix Form
6-Chocobo Form
7-Alexander Form
8-Shiva Form

And now here comes the explanation of each of the forms in order of acquisition:

Chocobo Form (6):
The first form Kairi gets in Twilight Town. Basically gives here a general enhancement of all her abilities and magics.
Shiva Form (8):
The second form Kairi gets in Agrabah, after defeating Shiva. Enhances her ice magic abilities, besides making her skate as if on ice.
Ifrit Form (3):
The third form Kairi gets in Olympus Colliseum, after defeating Ifrit. Increases her fire abilities and her attack attributes.
Ramuh Form (2):
The fourth form Kairi gets in Atlantis, after Ramuh decides to help her. Increases her electric abilities, and gives her the ability to 'thunder teleport'.
Diabolos Form (4):
The fifth form Kairi gets in Gallowmere, after defeting Diabolos. Offers her dark-type attributes and resistance to darkness, and gives her the ability to disguise as a Heartless, always she isn't hit by light sources.
Titan Form (1):
The sixth form Kairi gets in Troy, after saving Titan. Increases her physical strenght in great measure, and gives her earth-bassed power ups.
Alexander Form (7):
The seventh form Kairi gets in New York (Avengers Assemble), after saving Alexander. Increases her defense abruptly and gives her strong light-based attacks.
Esper/Phoenix Form (5)
The last form Kairi gets in the Phoenix Tower's ruins, but stabilized in Magix. The most powerful form Kairi can use, besides maturing and transforming her body.

Now everyone has a better idea of all the forms Kairi gets and uses in the story.