Deleting a board
Permanently deleting a board is not currently possible within Trello. However, it's possible to permanently delete cards, and to close boards.
Deleting information from a board
If the board contains sensitive information that you'd like deleted, the best thing to do is to delete the individual card containing the information. To do this, go into the card and select the More menu, then the Delete link. This will permanently destroy the information.
Closing a board
Close a board by selecting "Close Board" from the board menu (open the board's right sidebar and click the "Menu" button to expand the menu options). You have to be an admin of a board to close it.
Once a board is closed, it is no longer visible in your Boards list and can only be accessed by clicking the Boards button at the top left of your Trello page and selecting "See closed boards..."