Moderators oversee specific forums. They generally have the ability to edit and delete posts, move Questions, and perform other actions. Becoming a moderator for a specific forum is usually rewarded to users who are particularly helpful and knowledgeable in the subject of the forum 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 forums to create and how to organize them, what information to require from members and who to appoint as moderators.