iOS 10 has a lot of great new features and functionality for your iPad, iPhone, or iPod touch, but the new widget system is definitely a highlight. Almost all of the stock apps have widgets that offer timely information, and there are plenty of third-party apps too, which means we might finally be approaching parity with Android's widget selection.
Instead of just viewing widgets on the Today view, i.e., the main widgets screen, you can force-press an icon on your home screen to see that app's widget right then and there, above the regular Quick Actions, if any. This feature is only for stock Apple apps so far, but may appear in third-party ones if we're lucky.
If you like the widget enough, you can take things a step further by choosing the "Add Widget" option on this popup. This will add the app's widget to your Today view, which is accessible on the leftmost home screen, or by swiping to the left on your lock screen.
Normally, you'd have to go to the Today view, scroll down to the bottom, hit "Edit," and then add widgets that way, so this speeds things up considerably, though you'll still need to use the "Edit" page to reorganize the widgets layout.