If you're trying to get your site to post tweets to Twitter using the Twitter module and if, in the process of authorising the application, you end up with a message asking you to go back to your site and enter a provided PIN number, then you are a bloody idiot.
Just as I was.
The issue is that I did not follow instructions to enter a Callback URL as provided by my site at admin/config/services/twitter. To rectify this oversight, I entered the provided URL in the Twitter developer site's application in the appropriate field under the Settings tab. When this was done and the changes were saved, then I was able to authorise the application and all was well in my world once again.
Hope this helps, you idiot! :D