From a lengthy tutorial teaser piece:
We in the Windows team believe Action Center is more than simply a place to see all the notifications you’ve missed. We believe it is the place where the most relevant and timely content should show up, making it easy for you to take action on it. It should become your center of action.
As part of the Windows 10 Anniversary Update we undertook the challenge to make Action Center more relevant, beautiful, and personal than ever. This resulted in the introduction of several exciting new features – presented at //build – coupled with a visual update that focuses on highlighting more of the content that matters most to you.
...2. Download your favorite apps. Many of them already feature rich notifications, a few examples of these are: Flipboard, Skype UWP Preview, Facebook , LINE and many others!
3. Get web notifications from your favorite websites via Microsoft Edge. Open Microsoft Edge, visit a website that supports web notifications such as web.skype.com or web.groupme.com, sign-in, grant permission when prompted (only once), and as long as you don’t close the browser tab (it’s fine if tab is not in focus or not in foreground) you will get notifications directly in your Action Center
4. Get your phone’s notifications on your PC. Just make sure you are signed-in to Cortana on both devices with the same Microsoft Account, and you should start seeing your phone’s notifications mirrored onto your PC. It works for Android too, just download the latest Cortana app.
There's a lot more, including ways to control notifications in terms of specific applications or tabs, plus ways to get in touch with Action Center's architects, on the original, lengthy illustrated guide.
I've a feeling it may all be a little much for the average user, but maybe I'm just 'Windows Phone old-school'!
Note that some of the apps and services mentioned are not yet present in all regions, for example the new Skype UWP. Watch this space for more on this.