Wikipedia currently has a notification at the top of every single page saying that some accounts will be renamed due to a technical change when I am logged in. Clicking on Read more doesn't really explain much to me. What is it all about?
Wikipedia is currently trying to rename accounts that are currently not "global". As you know, the English Wikipedia is only one of the many wikis available on the network of wikis owned by the Wikimedia Foundation. Many years ago, they decided to implement something called "Single User Login (SUL)" so that users can just log in on one wiki and be logged in on every single wiki on the network. This is provided that an account is considered "global".
This should not affect many people, as most of our accounts are created after this change was implemented, which means your account may be global as well. You can visit Special:Preferences to check this. If not, you most likely have received a notification about this change personally sent to you, and your account will be renamed automatically so that it can become a global account.
Generally, you don't have to do anything, as this only affects a small number of users only. If you are one of these users, you would have to wait till August 2013 for your account to be automatically renamed before you can request for a proper name change on Meta. In the meantime, your username after this change would be in the form <user>~enwiki (if you have an account on the English Wikipedia.