Apple is walking back some of its restrictive response to EU regulations that have forced it to make changes to the iPhone and iOS. When the EU said that Apple must allow third-party web browsers on the iPhone, the company responded by cutting off web apps for the EU. After developers and some users complained, the company has changed its policy and will again allow EU users to save a website as an app on their home screen.
Web apps harken back to the original days of the iPhone, when there was no App Store. Instead, you could pin a web page to your home screen and it worked just like an app. The feature evolved to allow web apps to save data and send push notifications to the user. Macrumors has a good walkthrough of how web apps work and why they are a benefit. A web app is much smaller than an app that you download from the App Store, for instance.
