Forum - The forum is the whole installation, and usually contains multiple boards.
Board - A board contains topics, which in turn contain the posts. Boards can be private, so they are only visible to admins, moderators and members. Boards can be read-only, so they only allow posts by admins. Users can subscribe to a board, which means they will get all new posts in that board in one daily email.
Board with new posts
Topic - A topic, otherwise known as thread, contains all the posts on a specific subtopic, that should be named in the topic's subject. Boards have an expiration value that determines after how many days its topics will expire after their last post has been made. An important topic can be set to "protected", so that it never expires. Admins and moderators can lock a topic, so that no new posts can be made to it.
Topic with new posts
Hot topic with new posts
Post - A post is a single public message by a user. It can be either a first post, which starts a new topic, or a reply to an existing topic. Posts can be edited and deleted after posting.
User - A user is anyone who registers an account in the forum. Everyone needs to register to write in the forum, anonymous posting is not possible. However, there is no need for an account just for reading. Users can be granted membership status to selected boards, enabling them to see and post in private boards.
Forum Administrator - A forum admin can control and edit everything in the forum. A forum can have multiple forum admins.
Board Moderator - A board moderator's powers are limited to the boards he is admin of. A board moderators can edit, delete posts by normal users, mark topics as protected or locked, and delete topics. A board can have multiple moderators.