Threads belong to one board, posts belong to one thread. The post with parent_post = NULL is the first post in a thread, and is the one shown in boards view. All posts belonging to the same thread have the same parent_thread value. The server API returns posts as a flat view with parent_post pointing to the string id of the parent (all numeric ids are hidden from clients). The hierarchical thread view is built client-side.
Comments belong to one post. Chained comments are [x], are represented in the database as [y] and sent down to the client as [z]. Replies to comments.