The end notes will reveal the real names of the hearts, even though it's painfully obvious who's who. The main character is the Little Heart, and all characters are hearts themselves. In the kingdom hearts universe, there are three things that make up a person: Body, Soul, and Heart. This means, that they don't use gender pronouns such as "he, him" or "her, she", but "it, its", unless talking about their general person. (Lets say your heart is called Normal Heart, so it would be an it. If Normal Heart were to refer to you, Body, Soul and Heart as one, it would say "she/he failed epicly at volleying the volleyball")
And because Sora, Kairi, Riku (characters in KH) are all persons, their tale is ultimately different in some ways to their hearts own adventures. If that is confusing, just think about it like if Riku and Sora were talking, their hearts would also be thinking about the same conversation or sometimes a conversation that is closely related, but still different.
The hearts don't exist in the same black void, or body, (except Unwanted Heart and Little Heart for a while) but are mostly transparent looking to the other heart their talking to, kind of like a hologram, if the hearts choose to interact. (Like when Little Heart and another heart hug)
Since these Hearts are personified, they have "spiritual bodies", which you may or may not think as a sort of soul if it makes it easier, at which their center is the "core", or literal heart. Also, the Little Heart has an aura around it, which, if you've gotten a fairy sim before, is a "vibe" of emotions. It pretty much has a "happy aura" constantly going, which means that just being around it, makes you feel, well, happier! lol
K, just thought of the best idea ever, imagine that the hearts are like the emotions in Riley's head from Inside Out, but the Hearts only reside over the actual feeling of emotions, not the thoughts or actions the person does. Its pretty much like that lol
Also, to make sure there is no further confusion, the Soul is something that resides over personality, actions and more conscious thoughts. Which means, that in kingdom hearts 2, when (spoilers) Roxas and Sora battle it out in the World That Never was, in Sora's "heart" it was their Souls that we see. Sure, their hearts were there, but not from their exact point of view. You'll get it if you read it. Hopefully.
If you have any questions, feel free to comment them, and I hope you enjoy reading.
(Also, to those who want to listen to something at the same time, I'm working on a music compilation, but until then, I suggest you listen to each characters theme parallel to when said character is mentioned. Sets the mood, you know?)