"Vanity" URLs have been available to some high-ranking Google+ users for a while, but now Google is dramatically expanding the system. This means you will likely be able to customize your account page with a more memorable address. Instead of a bunch of meaningless numbers, users' URLs now look like: http://plus.google.com/+Username.
Google is rolling out the system to users right now. To get one, you must verify your Google account with a phone number, have had an account for at least a month, and have at least 10 followers.
I think this is a very big move for Google. The vanity URL makes it easier for potential followers to find you on Google+. More significantly, it makes the Google+ network a more human place. Owning your Plus homepage is now a more touchy-feely experience because, for example, to share your account details you don't have to email someone a stupid machine-like string of numbers to tell them the address of your Plus profile. It's also the sort of label that can go on a business card or in an email footer. This trick is something Twitter got right at the start, and even Facebook has caught up. The big question is, what on Earth took Google so long?