Party chat is where you can have multiple people in a chat, talking, at the same time. Private chat is where it is only you and one other person. This makes it more private for you and the other person as you can not have more then two people in the chat.
This will be between you and a friend only.
As for why this feature is not supported on the Xbox One. In short, I don't know. This could be because of how many people used the feature or how the feature worked. If you would like this feature on Xbox One, then you can make a suggestion on the Xbox