Double opt-in is when a person confirms that they did indeed sign-up as a subscriber to your mailing list not once but twice. The first opt-in occurs when they go to your web site and enter their email address to recieve your newsletter. They are then sent an email to confirm their subscription. When they have confirmed their subscripton using both methods the user is considered a double opt-in. Current Can-Spam rules dictate that all mailing list subscribers must be double opt-in.
A double opt-out is when a person confirms they wish to unsubscribe from your list on two occasions. The first opt-out occurrs when they go to your website and enter their username or email address to unsubscribe. They are then sent an email which contains a link that they must click to confirm that they wish to unsubscribe. When a person has confirmed their unsubscription using both method they are considered a double opt-out.