Moderators and Administrators generally have the ability to edit and delete posts, move threads, and perform other actions. Becoming a moderator for a specific website is usually rewarded to users who are particularly helpful and knowledgeable in the subject of the website they are moderating.
Administrators are the people who have overall control of everything that happens on the board. They oversee how the board is styled, what website to create and how to organize them, what information to require from members and who to appoint as moderators.