The below process outlines how to delete a message thread between yourself and another user:
Navigate to your Messages, by clicking the envelope icon located towards the top right side of the page
Go to the message thread in question that you want to delete and click the Delete text link in the upper right-hand corner of the message window.
A pop-up window will appear and you will be asked to confirm the deletion by selecting the blue Delete button again. After clicking the "Delete" button, the entire message will be deleted and unrecoverable.