As an admin, if a connection to one of your social channels fails (ex: a Facebook password change) and requires reauthorization, you will be notified both via email and within your recent activity feed. You will also see a warning icon within the Integration Admin interface.
From the Integration Admin, click the channel that requires reauthorization and then click the Re-Authorize link in the upper right.
This will trigger an authorization form that will require you to input the user id and password associated with that social channel. Proceed through the authorization process and if successful, you should be good to start posting content to that channel.